Problems in Mathematical Analysis I: Real Numbers, Sequences and Series [1 ed.] 0821820508, 9780821820506

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http://dx.doi.org/10.1090/stml/004

Selected Titles in This Series Volume 4 W . J. Kaczor and M. T. Nowak Problems in mathematical analysis I: Real numbers, sequences and series 2000 3 Roger Knobel An introduction to the mathematical theory of waves 2000 2 Gregory F. Lawler and Lester N . Coyle Lectures on contemporary probability 1999 1 Charles Radin Miles of tiles 1999

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Problems in Mathematical Analysis I Real Numbers, Sequences and Series

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STUDENT MATHEMATICAL LIBRARY Volume 4

Problems in Mathematical Analysis I Real Numbers, Sequences and Series W J. Kaczor M.T. Nowak

iAMS

AMERICAN MATHEMATICAL SOCIETY

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Editorial Board David Bressoud R o b e r t Devaney, Chair

Carl Pomerance Hung-Hsi W u

Originally published in Polish, as Zadania z Analizy M a t e m a t y c z n e j . Cz$sc Pierwsza. Liczby Rzeczywiste, Ciagi i Szeregi Liczbowe © 1996, W y d a w n i c t w o U n i w e r s y t e t u Marii Curie-Sklodowskiej, Lublin. Translated, revised a n d a u g m e n t e d by t h e a u t h o r s . 2000 Mathematics

Subject Classification.

Primary 00A07, 4 0 - 0 1 .

Library of Congress C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Kaczor, W. J. (Wieslawa J.), 1949[Zadania z analizy matematycznej. English] Problems in mathematical analysis. I. Real numbers, sequences and series / W. J. Kaczor, M. T. Nowak. p. cm. — (Student mathematical library, ISSN 1520-9121 ; v. 4) Includes bibliographical references. ISBN 0-8218-2050-8 (softcover : alk. paper) 1. Mathematical analysis. I. Nowak, M. T. (Maria T.), 1951- II. Title. III. Series. QA300K32513 2000 515 / .076-dc21 99-087039 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org. © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2

09

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Contents

Preface Notation and Terminology Problems Chapter 1. Real Numbers 1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions 1.2. Some Elementary Inequalities Chapter 2. Sequences of Real Numbers 2.1. Monotonic Sequences 2.2. Limits. Properties of Convergent Sequences 2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications 2.4. Limit Points. Limit Superior and Limit Inferior 2.5. Miscellaneous Problems Chapter 3. Series of Real Numbers 3.1. Summation of Series

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Contents

Vlll

3.2. Series of Nonnegative Terms

72

3.3. The Integral Test

88

3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz

92

3.5. The Dirichlet and Abel Tests

99

3.6. Cauchy Product of Infinite Series

102

3.7. Rearrangement of Series. Double Series

105

3.8. Infinite Products

112

Solutions Chapter 1. Real Numbers 1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions

125

1.2. Some Elementary Inequalities

136

Chapter 2. Sequences of Real Numbers 2.1. Monotonic Sequences

151

2.2. Limits. Properties of Convergent Sequences

162

2.3. The Toeplitz Transformation, the Stolz Theorem and their Applications

181

2.4. Limit Points. Limit Superior and Limit Inferior

189

2.5. Miscellaneous Problems

208

Chapter 3. Series of Real Numbers 3.1. Summation of Series

245

3.2. Series of Nonnegative Terms

269

3.3. The Integral Test

302

3.4. Series of Positive and Negative Terms - Convergence, Absolute Convergence. Theorem of Leibniz

309

3.5. The Dirichlet and Abel Tests

324

3.6. Cauchy Product of Infinite Series

333

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Contents

IX

3.7. Rearrangement of Series. Double Series

342

3.8. Infinite Products

360

Bibliography - Books

379

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Preface

This book is an enlarged and revised English edition of a Polish version published in 1996 by the Publishing House of Maria CurieSklodowska University in Lublin, Poland. It is the first volume of a planned series of books of problems in mathematical analysis. The second volume, already published in Polish, is under translation into English. The series is mainly intended for students who take courses in basic principles of analysis. The choice and arrangement of the material make it suitable for self-study, and instructors may find it useful as an aid in organizing tutorials and seminars. This volume covers three topics: real numbers, sequences, and series. It does not contain problems concerning metric and topological spaces, which we intend to present in subsequent volumes. The book is divided into two parts. The first part is a collection of exercises and problems, and the second contains their solutions. Complete solutions are given in most cases. Where no difficulties could be expected or when an analogous problem has already been solved, only a hint or simply an answer is given. Very often various solutions of a given problem are possible; we present here only one, hoping students themselves will find others.

XI

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xi i

Preface

With the student in mind, we have tried to keep things at an elementary level whenever possible. For example, we present an elementary proof of the Toeplitz theorem about the so-called regular transformation of sequences, which in many texts is proved by methods of functional analysis. The proof presented is taken from Toeplitz's original paper, published in 1911 in Prace Matematyczno-Fizyczne, Vol. 22. We hope that our presentation of this part of real analysis will be more accessible to readers and will ensure wider understanding. All the notations and definitions used in this volume are standard and commonly used. The reader can find them, for example, in the textbooks [12] and [23], in which all necessary theoretical background can be found. However, to make the book consistent and to avoid ambiguity, a list of notations and definitions is included. We have borrowed freely from many textbooks, problem books and problem sections of journals like the American Mathematical Monthly, Mathematics Today (Russian) and Delta (Polish). A complete list is given in the bibliography. It was beyond the authors' scope to trace all original sources, and we may have overlooked some contributions. If this has happened, we offer our sincere apologies. We are deeply indebted to all our friends and colleagues from the Department of Mathematics of Maria Curie-Sklodowska University who offered stimulating suggestions. We have had many fruitful conversations with M. Koter-Morgowska, T.Kuczumow, W. Rzymowski, S. Stachura and W. Zygmunt. Our sincere thanks are also due to Professor Jan Krzyz for his help in preparing the first version of the English manuscript. We are pleased to express our gratitude to Professor Kazimierz Goebel for his encouragement and active interest in the project. It is our pleasure to thank Professor Richard J. Libera, University of Delaware, for his invaluable and most generous help with the English translation and for all his suggestions and corrections which greatly improved the final version of the book. W. J. Kaczor, M. T. Nowak

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Notation and Terminology

• • • • • • • • •

R - the set of all real numbers R + - the set of all positive real numbers Z - the set of all integers N - the set of all positive integers Q - the set of all rationals (a, b) - open interval with the endpoints a and b [a, b] - closed interval with the endpoints a and b [x] - the integral part of a real number x For x e R, ( 1 sgnx = < - 1 0

for x > 0, for x < 0, for x = 0.

For n e N, n! = 1 - 2 - 3 - . . . - n , (2n)!! = 2 • 4 • 6 •... • (2n - 2)(2n) and (2n - 1)!! = 1 • 3 • 5 • ... • (2n - 3)(2n - 1). If A c K is nonempty and bounded from above, then sup A denotes the least upper bound of A. If a nonempty set A is not bounded above, then we assume that sup A = -foo. xin

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Notation and Terminology

XIV

If A c t is nonempty and bounded from below, then inf A denotes the greatest lower bound of A. If a nonempty set A is not bounded below, then we assume that inf A = — oo. A sequence {an} of real numbers is said to be monotonically increasing (monotonically decreasing) if a n + i > an for all n £ N (a n +i < an for all n £ N). The class of monotonic sequences consists of the increasing and the decreasing sequences. A number c is a limit point of the sequence {an} if there is a subsequence {ank} of {an} converging to c. Let S be the set of all the limit points of {an}. The limit inferior, lim a n , and the limit superior, lim a n , of the sequence {an} are defined as follows: +oc lim an = < —oo

lim an

n—+oc

if {an} is not bounded above, if {an} is bounded above and S = 0,

supS

if {an} is bounded above and S ^ 0,

-oo

if {an} is not bounded below,

-f oo

if {an} is bounded below and S = 0,

inf S

if {an} is bounded below and S ^ 0.

An infinite product

Yl an is said to be convergent if there 71=1

exists no £ N such that an ^ 0 for n > UQ and the sequence {a n o a n o + i • ... • anQ+n} converges, as n —> oo, to a limit P0 other than zero. The number P = a ^ • • • • * a>n0-i ' Po is called the value of the infinite product.

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Problems

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Chapter 1

http://dx.doi.org/10.1090/stml/004/01

Real Numbers

1.1. Supremum and Infimum of Sets of Real Numbers. Continued Fractions 1.1.1. Show that sup{x e Q : x > 0 , x2 < 2} = V2. 1.1.2. Let A C M be a nonempty set. Define —A = {x : —x e A } . Show that sup(—A) = —inf A, inf (—A) = - s u p A. 1.1.3. Let A, B e l

be nonempty. Define

A + B = {z = x + y:xeA,

yeB},

A-B

y €B}.

= {z = x-y:xeA,

Show that sup(A -f B) = sup A + sup B , sup( A - B) = sup A - inf B. Establish analogous formulas for inf (A + B) and inf (A — B). 3

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Problems. 1: Real Numbers

4

1.1.4. Given nonempty subsets A and B of positive real numbers, define A • B = {z = x - y : x € A, y eB}

A = {Z = ± : X € A } . Show that sup(A • B) = sup A • sup B . Show also that, if inf A > 0, then

sup

(1) = ex

and, if inf A = 0, then sup (^-) = +oo. Additionally, show that if A and B are bounded sets of real numbers, then sup(A • B) = max{sup A • sup B , sup A • inf B , inf A • sup B , inf A • inf B } . 1.1.5. Let A and B be nonempty subsets of real numbers. Show that sup(A U B) = max{sup A, sup B} and inf (A U B) = min{inf A, inf B } . 1.1.6. Find the least upper bound and the greatest lower bound of A i , A2 defined by setting

Ai = J2(-l)"+1 + ( - 1 ) ^ (2 + £) : n €N} , fn-1 2717T 1 A2 = < 7 cos —— : n e N > .

\n+l

3

J

1.1.7. Find the supremum and the infimum of the sets A and B , where A = {0.2,0.22,0.222,... } and B is the set of decimal fractions between 0 and 1 whose only digits are zeros and ones. 1.1.8. Find the greatest lower and the least upper bounds of the set of numbers \n^' , where n e N.

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1.1. Supremum and Infimum. Continued Fractions

5

1.1.9. Find the supremum and the infimum of the set of numbers n+ 2~l 2"m , where n,ra G 1.1.10. Determine the least upper and the greatest lower bounds of the following sets: (a)

A = < — : m, n G N, m < 2n \ ,

(b)

B = { v ^ ~ [y/n] : n G N} .

1.1.11. Find (a)

sup{a: e R : x 2 + a ; + l > 0 } ,

(b)

mf{z = x + x~1 : x > 0},

(c)

inf{2 = 2* + 2-

:x>0}.

1.1.12. Find the supremum and the infimum of the following sets: (a)

A = < — + — : m, n G N > , [ n m J

(c)

C

(d)

D = i T — ^ — : m G Z, n G N i , ^ |m| -h n. J

(e)

E= 4

\ m : ra,neN>, [m + n J

f

^^

T,T1

: m, n G N > .

1.1.13. Let n > 3 be an arbitrarily fixed integer. Take all the possible finite sequences ( a i , . . . , a n ) of positive numbers. Find the least upper and the greatest lower bounds of the set of numbers

E: ^ afc + afc

Ofc +1

-f afc+2

where we put a n +i = a\ and a n +2 = &2 •

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Problems. 1: Real Numbers

6

1.1.14. Show that for any irrational number a and for any positive integer n there exist a positive integer qn and an integer pn such that _Pn\ _1_ qn I nQn Show also that {pn} and {qn} can be chosen in such a way that we have a—


1 + ai + a 2 + ... + an .

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9

1.2. Some Elementary Inequalities

Remark. Note that if a\ = a 2 = ... = an = a then we get the well known Bernoulli inequality: (1 + a)n > 1 + n a , a > - 1 . 1.2.2. Using induction, prove the following result: If a i , a 2 , . . . , a n are positive real numbers such that a\ • a 2 •... • an = 1, then ai + a 2 + ... + an>n. 1.2.3. Let An, Gn and Hn denote the arithmetic, geometric and harmonic means of n positive real numbers a i , a 2 , ...,a n ; that is, _ Q>1 + Q>2 + ... + CLn

A

An — Gn

= 71

,

n

\/0>l ' . . . • CLn j

"" i + i + ... + -^ *

Show that An > Gn > Hn. 1.2.4. Using the result (Gn < An) in the foregoing problem, establish the Bernoulli inequality (1 + x)n > 1 + nx for x > 0. 1.2.5. For n €N, verify the following claims: 1

1

1

1

2

(b)

^TI + ^T2 + ^T3 + - + 3 ^ T T > 1 '

W

1 1 < 2 3n + l

(d)

n(v / ^TT-l) 2, show that

/ 2 „_ 2 y-i fc=0

x

'

1.2.37. Let a,k > 0, fc = 1,2, ...,n, and let A n be their arithmetic mean. Show that for any integer p > 1,

fc=i

^

fc=i

1.2.38. For positive a^, fc = 1,2,..., n, we set a = ai + c&2 + ... -f a n . Show that n—1

2

fc=l

1.2.39. Show that for any rearrangement 6i, 6 2 ,..., 6n of the positive numbers oi,a2, ...,o n ,

T-n'

r + h~ + "' +

1.2.40. Prove the Weierstrass inequalities: If 0 < a& < 1, fc = 1,2,..., n, and a\ + c&2 + ... + a n < 1, then n

(a)

n

1

1

afc

1 + J2 * < I I t + ) < /c=i n

(b)

a

fc=i

n

..

1 fl

1 - J2 "* < II^ - *)< /s=i

n >

1 — ^ a^ fc=i

fc=i

n •

i + J2 ak fc=i

1.2.41. Assume that 0 < a^ < 1, fc = 1,2, ...,n, and set ai 4- a2 + ... + o n = o. Show that ak

Err a

fc

na n-a

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Problems. 1: Real Numbers

16

1.2.42. Let 0 < Ofc < 1, fc = 1,2,..., n, and n > 2. Verify the inequality 71

n

n Y, ak fc=i

Y-±—< — Z-f1 1 ++ aa* - « 1

fc=i

*

Eajb + nfta* fc=i fc=i

1.2.43. For nonnegative a^, A; = 1,2, ...,n, such that a\ + 0,2 + ... 4a n = 1, show that

n(1 + afc )^( n+1 ) n II afc '

(a)

n

n

n^ - *)^^- )"!!0*-

(b)

1 0

1

fc=l fc=l

1.2.44. Show that if a& > 0, A; = 1,2, ...,n, and ^

k=l-'lk

then

j ^ ^ - = n — 1,

ni-> ( n-ir.

k=lak

1.2.45. Prove that under the assumptions of 1.2.43, we have n nu+afc)

na-flfc)

^ (n r— n > ^r—vx— + ^l ) — " (n-l)n >

n>l.

1.2.46. Show that for positive ai, 02,..., a n , a i

02 + ^3

1

a

2

03+^4

,

,

a

n-2

an_i+an

fln-1

an-hai

,

Qn

a\ + 02

. ™

4

1.2.47. Let £ and a i , a 2 , . . . , a n be any real numbers. Establish the inequality

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1.2. Some Elementary Inequalities

17

1.2.48. Show that for positive ai,a2, ...,an> and 61,62, ...,6 n , we have y/fa

+ 6i)(a 2 + 6 2 )...(a n + 6n) > ^aia 2 ...a n + ^ i ^ . . . 6 n .

1.2.49. Assume that 0 < a\ < 0,2 < ... < a n and Pi,P2>---?Pn are n

nonnegative such that Yl Vk = 1- Establish the inequality

where A = ^(ai + a n ) and G = ^/ala^. 1.2.50. For a positive integer n, let cr(n) and r(n) denote the sum of all the positive divisors of n and the number of these divisors, respectively. Show that ^ y > y/n.

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Chapter 2

http://dx.doi.org/10.1090/stml/004/02

Sequences of Real Numbers

2.1. Monotonic Sequences 2.1.1. Show that (a) if a sequence {an} is monotonically increasing, then lim an = n—>oc

sup{a n : n G N}; (b) if a sequence {an} is monotonically decreasing, then lim an = inf{a n : n G N}.

n—>oo

2.1.2. Let ai,a2,...,a p be fixed positive numbers. Consider the sequences a7t + a2 + ... + a%

and xn = \fs^ , n G P Show that the sequence {xn} is monotonically increasing. Hint. First establish monotonicity of the sequence -J -^- >, n > 2. 2.1.3. Show that the sequence {a n }, where an = ~ , n > 1, strictly decreases and find its limit. 2.1.4. Let {an} be a bounded sequence which satisfies the condition o>n+i > o n — T^r, n G N. Show that the sequence {a n } is convergent. Hint. Consider the sequence {an — ^r=r} • 19

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Problems. 2: Sequences of Real Numbers

20

2.1.5. Prove the convergence of the sequences:

(a)

an = _ 2 ^ + (i ! +

-L + ... + -L);

(b)

6n = - 2 v ^ + ( ^ + - l +

... + -I=).

Hint. First establish the inequalities 1 2(y/n + 1 - 1) < - = + -7= + ... + - = < 2Vn, Vl v2 yn

n €N.

2.1.6. Show that the sequence {an} defined recursively by a\ = - ,

a n = -y/3an_i — 2 for n > 2,

converges and find its limit. 2.1.7. For c > 2, define the sequence {an} recursively as follows: a\ = c2,

a n +i = (a n - c) 2 ,

n > 1.

Show that the sequence {an} strictly increases. 2.1.8. Suppose that the sequence {an} satisfies the conditions 0 < an < 1,

an(l - an+i) > ~ for n G N.

Establish the convergence of the sequence and find its limit. 2.1.9. Establish the convergence and find the limit of the sequence defined by &i = 0, a n + i = \/6 -I- an for n > 1. 2.1.10. Show that the sequence defined by ai=0,

a2=2>

an+i = g ( l + an + an-i)

for

n > 1

converges and determine its limit.

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2.1. M o n o t o n i c Sequences

21

2.1.11. Study the monotonicity of the sequence a

H h

"=(2^TIJ!!'

-

and determine its limit. 2.1.12. Determine the convergence or divergence of the sequence (2n)!! . (2n+l)!!

n> 1.

2.1.13. Prove the convergence of the sequences (a)

(b)

0

« =

1 +

22

+

3 2 + - + ^>

i + J_ + J_ + ... + J_,

Gn =

n e N

5

neK

2.1.14. Show the convergence of the sequence {a n }, where 1 1 1 an — —====. -\ . -f ... H , =, y/n(n + 1) ^/(2n - l)2n x / ( n - h l ) ( n + 2)

n G N.

2.1.15. For p G N , a > 0 and ai > 0, define the sequence {an} by setting a n + i = - ((p - l)an + - ^ r r J > ^ G N. Determine lim a n . n—•oo

2.1.16. Define {a n } recursively by a n + i = J 2 -f >/^n

ax =

for n > 1.

Prove the convergence of the sequence {an} and find its limit. 2.1.17. Define the recursive sequence {an} as follows: 2(2a n + 1 ) ^_^ —— for n G N. an + o Establish the convergence of the sequence {an} and find its limit. 1

ai = 1, a n +i =

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Problems. 2: Sequences of Real Numbers

22

2.1.18. Determine all c > 0 such that the recursive sequence {an} defined by setting 1

c a

i = 2'

a

n+i==2^C+

a

^

f o r n

^

N

converges. In case of convergence find lim an. n—•oo

2.1.19. Let a > 0 be fixed and define the sequence {an} by setting a ^ i n + 3° a\ > 0 and a n + i = OLnTr-^

Sal + a

_ __ lor n G R

Determine all a\ for which the sequence converges and in such a case find its limit. 2.1.20. Let {an} be defined recursively by Gn+i = ~A ^— for n > 1. 4 - 6an Determine for which a\ the sequence converges and in case of convergence find its limit. 2.1.21. Let a be arbitrarily fixed and let {an} be defined as follows: a\ £ R

and

a n + i = a\ + (1 — 2a)an -f a 2

for

n G N.

Determine all ai such that the sequence converges and in such a case find its limit. 2.1.22. For c > 0 and 6 > a > 0, define the recursive sequence {an} by setting

al + ab

^

a\ = c, a n + i = —-—— for n e N. a-\-b Determine for which values a, b and c the sequence converges and find its limit. 2.1.23. Prove the convergence and find the limit of the sequence {an} defined inductively by a\ > 0,

an+i = 6-, 7 + an

n € N.

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23

2.1. Monotonic Sequences 2.1.24. For c > 0, define the sequence {an} as follows: ai = 0, an+\ = y/c + an, n e N. Show the convergence of the sequence and determine its limit. 2.1.25. Investigate the convergence of the sequence defined by a\ = y/2, an+\ = \f2a~n~, n e N.

2.1.26. Let k E N be fixed. Study the convergence of the sequence {an} defined by setting a1 = >/5, a n + i = >/5a^, n e 2.1.27. Investigate the convergence of the sequence {a n } given by 1 < ai < 2,

a2n+1 = 3a n - 2, n G N.

2.1.28. For c > 1, define the sequences {an} and {6n} as follows: (a)

ai = \Jc{c-

(b)

6i = \/c,

1),

a n + i = y/c(c-

1) + a n , n > 1;

&n+i = \/c6^, n > 1.

Prove that both sequences tend to c. 2.1.29. Given a > 0 and 6 > 0, define the sequence {an} by setting 0 < ai < 6, Find

a n + i = A/ -^ V a 4-1

for

n > 1.

lim a n .

n—+oo

2.1.30. Prove the convergence of {an} defined inductively by a\ = 2, and find its limit.

a n +i =2-1- ——— 3

+f

for

n > 1

On

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Problems. 2: Sequences of Real Numbers

24

2.1.31. The recursive sequence {an} is given by setting a\ = 1,

a>2 = 2,

a n + i = yJan-\ + yfa^

for

n > 2.

Show that the sequence is bounded and strictly increasing. Find its limit. 2.1.32. The recursive sequence {an} is given by setting ai = 9,

a2 = 6,

an+\ = y/an-i

+ ^Ja^

for

n > 2.

Show that the sequence is bounded and strictly decreasing. Find its limit. 2.1.33. Define the sequences {an} and {bn} as follows: 0 0, set an = 2 y / x, n G N. Show that the sequence {an} is bounded. Show also that it is strictly increasing if x < 1 and strictly decreasing if x > 1. Compute lim an. Moreover, put cn = 2 n ( a n - l )

dn = 2n (1 - — )

and

Show that {c n } is decreasing and {dn} quences have the same limit.

for

n G N.

is increasing and both se-

2.2. Limits. Properties of Convergent Sequences 2.2.1. Calculate: (a)

lim \ / l 2 + 2 2 + ... + n 2 ,

(b)

lim

(c) (d) (e)

n + sin n 2

n-* n + cos n n—>oc

lim n—>oc

l - 2 + 3 - 4 + . . . + (-2n) /

9

. ,

lim (V2 - ^2)(>/2 - #2) •... • (V2 n lim _,

2

"V2),

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2.2.

Limits. P r o p e r t i e s of Convergent Sequences

27

71'

- . , ,. / 1 2 n ( h ) n->oo „ l l n L y nr ¥2 -+T T1 + Z - + n2 + n n 2T -+T 2^ + '*

'

/.x / n 2n nn W n™oo _ h l ?_ V -n T3 T + T1 + Z-T^T n 3 -f 2 + -' " + n 3 -f n 2 . 2 . 2 . Let 5 > 0 and p > 0. Show t h a t hm — n->oo (1 2 . 2 . 3 . For a G (0,1), calculate 2 . 2 . 4 . For a £ Q, calculate 2 . 2 . 5 . Show t h a t the limit

— = 0. +p)n lim ((n + l ) a n—•oo

na).

lim sin(n!a7r). n—>-oo lim s i n n does not exist. n—>oo

2 . 2 . 6 . Show t h a t for any irrational a the limit not exist.

lim sinna7r does n—>-oo

2 . 2 . 7 . For a € R, calculate

hm -

n->oo n VV

a+-

n/

+U+\

nj

2 . 2 . 8 . Suppose an ^ 1 for all n and integer A:, c o m p u t e lim

n—>oo

+ ... + a + \

n

lim an = 1. Given a positive

71—>00

an + QTl + ... + a ^ - f c ^ an — 1

2 . 2 . 9 . Find ,. 1 1 I N f" h- m T-ir-z + TT-z—: + + ooVl-2-3 2-3-4 "' n-(n + l)-(n + 2 ) y

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Problems. 2: Sequences of Real Numbers

28 2.2.10. Calculate

-=-fc3-l hm TT 7-5—-. 3

n-^00 1 1 fc + 1 k=2

2.2.11. Determine

lim n

i

EE n3

3

t=i j=i

2.2.12. Compute

-3 J-4 I •... • ( 1 -

lim [ 1 - A)(l

n->oc\

2-3/ V

* /

V

(n + l ) - ( n + 2)

2.2.13. Calculate ,.

hm n

A f c 3 + 6A;2 + llfc + 5 >

TZ

—

.

(fc + 3 ) !

^°°k^i

2.2.14. For x ^ - 1 and x ^ 1, find n

lim

El

fc=l

2fc_1

2="-

2.2.15. Determine for which x G R the limit lim 1 7 ( 1 + x 2 * ) -1"-1"

l—•OO n—•oo

exists and find its value. 2.2.16. Determine all x G R such that the limit n

,

lim fc=Q 17 Ix 1 + S

n—•oo

A A

\

2 t l

exists and find its value. 2.2.17. Establish for which x G R the limit

lim f](l + x*k +x™k) n—• oo

A l

fc=l

exists and find its value.

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2.2. Limits. Properties of Convergent Sequences

29

2.2.18. Calculate ,. l - l ! + 2-2! + ... + n - n ! hm — .

n-+oo

(n+1)!

2.2.19. For which x E M does the equality n 1999

j

lim n-oo nx - (n - l ) x

2000

hold? 2.2.20. Given a and b such that a > 6 > 0, define the sequence {an} by setting ab a\ = a -f 6, a n = oi , n > 2. Gn-l

Determine the nth term of the sequence and compute lim an. n—•oo

2.2.21. Define the sequence {an} by setting (H = 0,

a2 = 1 and

a n + i — 2a n + a n _i = 2

for

n > 2.

Determine its nth term and calculate lim o n . n—+oo

2.2.22. For a > 0 and 6 > 0, consider the sequence {a n } defined by ai =

a6 y

an =

: and =,

n > 2.

Determine its nth term and find lim an. n—>oo

2.2.23. Let {a n } be a recursive sequence defined as follows: ai = 0,

an =

an_i+3

,

n > 2.

Find the formula for the n t h term of the sequence and find its limit. 2.2.24. Study the convergence of the sequence given by a\ = a,

a n = 1 + 6o n _i,

n > 2.

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Problems. 2: Sequences of Real Numbers

30

2.2.25. The Fibonacci sequence {an} is defined as follows: a\ = d2 = 1,

fln+2

= 0"n + G n + l ,

Tl > 1.

Show that an = a-0 ' where a and /? are roots of x2 = x + 1. Compute lim tfa^. n—+oo

2.2.26. Define the sequences {an} and {bn} by setting a\ = a, a n + bn

h = 6, bn +1

Show that lim an = lim 6 n . n—+00

n—>oo

2.2.27. Given a £ {1,2,...,9}, compute n digits

a 4- aa -f ... + aa...a lim 10 s ' n—>oo 2.2.28. Calculate

lim (tfn- l) r

2.2.29. Suppose that the sequence {an} converges to zero. Find lim a". n-+oo

n

2.2.30. Given positive Pi, P2> ••• >Pfc a n d a i , a 2 ,... ,a^, find lim n^oo

PiaxT +p2a2 + — +Pk% pia^ + p 2 a£ + ... + p^a^

2.2.31. Suppose that lim

Qn+1

q. Show that

n—>oc

(a) if q < 1, then lim a n = 0, n—>oo

(b) if (j > 1, then lim |a n | = oc.

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2.2. Limits. Properties of Convergent Sequences

31

2.2.32. Suppose that lim y/\an\ — q. Show that n—>oo

(a) if q < 1, then lim an = 0, n—• oo

(b) if q > 1, then lim |a n | = oo. n—KX>

2.2.33. Given a real number a and x G (0,1), calculate lim

naxn.

n—•oc

2.2.34. Calculate m(m - 1) • ... • (m - n + 1) nn hm —-^ -x ,

n—*oo

n\

2.2.35. Assume that Show that

lim anbn

. for

__ , . , , m G N and |z| < 1.

lim an = 0 and {6 n } is a bounded sequence.

n—KX>

=0.

n—>oo

2.2.36. Show that if lim an = a and lim bn = 6, then n—>oo

n—•oo

lim max{a n , 6 n } = max{a, b}.

n—>oo

2.2.37. Let an > - 1 for n G N and let lim a n = 0. For p G N, n—KX>

find lim ^/l -f- an.

n—+oo

2.2.38. Assume that a positive sequence {an} For natural p > 2, determine lim n->oo

^ ^ ~

1

converges to zero.

,

a„

2.2.39. For positive ai,a2, ...,a p , find lim f y (n + ai)(n + a 2 ) •... • (n + a p ) - n J.

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Problems. 2: Sequences of Real Numbers

32

2.2.40. Calculate 1 ,. ( 1 1 \ lim ,0 4- , , + ... 4- - = = = = n-oo V>/n2 + 1 V V T 2 \/n2+n + l/ 2.2.41. For positive ai,ei2, ...,a p , find

lim

n/a?+aJ

+ ... + aJ

2.2.42. Compute n 1999

n 1999

lim \ / 2 sin2 4- cos2 n->oc V n -f 1 n -f 1 2.2.43. Find lim (n 4- 1 4- n cos n) 2n+nsin? 2.2.44. Calculate

sz,±{f^-i

2.2.45. Determine

*SX(\KJ-I 2.2.46. For positive a^, fc = 1,2, ...,p, find p

lim

- Y" ^afc

2.2.47. Given a € (0,1), compute n-l

lim Y ] ( a + - )

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2.2. Limits. Properties of Convergent Sequences

33

2.2.48. Given real x > 1, show that lim (2^/x-l)n

= x2.

2.2.49. Show that

Km e ^ : 1 ) " = i. 2.2.50. Which of the following sequences are Cauchy sequences? .

(a)

tanl

tan 2

tann

2

2n

an = —— + —=+ ... + 2 ,

1

2

, 1 1 2 + 3+

= 1+

1 -+n'

(c)

a

a

(e)

an = a i ^ 1 + a 2 g 2 + ... + an 0 ' n

for each fc G N,

n—>oo

(ii)

y^c n ) / c —> 1,

(iii)

there exists C > 0 such that for all positive integers n:

*-—* fc=l

ro—>oo

n

^|cn,fc|k, n> 1, is also convergent andn lim 6 n = fc=i ' ~"°° lim an.

n—>oo

2.3.2. Show that if lim an = a, then n—->oo

ai + a 2 + ... + a n

lim

n—>oo

n

= a.

2.3.3. (a) Show that the assumption (iii) in the Toeplitz theorem (Problem 2.3.1) can be omitted if all the numbers cn^ are nonnegative. (b) Let {bn} be the transformed sequence defined in the Toeplitz theorem with cn^ > 0, 1 < fc < n, ra > 1. Show that if lim an = n—>oo

-|-oo, then lim bn = +oo. n—KX)

2.3.4. Show that if lim an — -hoc, then n—>oo

lim

n—>oo

oi + a 2 + ...+ a n 77,

= -foo.

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36

P r o b l e m s . 2: S e q u e n c e s o f R e a l N u m b e r s

2 . 3 . 5 . Prove t h a t if lim an = a, then n—»oo lim

na\ -f (n — l)a2 + ... + 1 • an

a

n—>oo

2.3.6. Show t h a t if a positive sequence {an} lim i / a i •... • a n = a.

converges t o a, then

n—•oo

2 . 3 . 7 . For a positive sequence { a n } , show t h a t if then

lim ^ ^

= a,

lim r/a^, — a. n—>-oo

2.3.8. Let

lim a n = a and

n—>oc

r

lim

2.3.9. Let {an} (i) (ii)

^ i 6 n + « 2 ^ n - l + ... +Clnbi

and {bn}

6 n > 0, n e N, lim

lim 6 n = 6. Show t h a t

n—+00

and

= ab.

be two sequences such t h a t lim ( 6 i + b 2 + ... + &„) =+cx>,

-^=9-

Prove t h a t ,. a i + a2 + ... + an lim —- = g. n-+oo b\ + 02 + ... "f 0 n 2 . 3 . 1 0 . Let { a n } and {6 n } be two sequences for which (i)

bn > 0, n € N,

(ii)

lim a n = a.

and

lim (&i -f b2 + ... 4- 6 n ) = + o o ,

n—•oo

Show t h a t lim n-^oo

ai&i + a20 2 + ... + anbn = a. 6i + D2 + ... + 0 n

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2.3. Toeplitz Transformation a n d Stolz Theorem

37

2.3.11. Using the result in the foregoing problem, prove the Stolz theorem. Let {xn},

{yn} be two sequences that satisfy the conditions:

(i)

{yn}

(n) n

strictly increases to + oo,

lim = g. -*°° 2/n - 2/n-l

Then n—oo yn 2.3.12. Calculate (a)

lim -i= M + ^ + ... + - L ) ,

(b)

lim _

(c)

lim -T-T-

(d) (e) (f) (g)

—- [a + — + ... + —

n—*oo n

a > 1, (fc + n)!

1!

^

W —F= 1 + lim -7=

1 /

n!

+ ...+

/ceN,

1

.. lfc + 2fc + ... + n* , ^T lim — , k e N, fc+i rv 1 + 1 - a + 2- a 2 + ... + n - a n , a > 1, lim n—>oo n • an+1 lim

n—>oc

nfc v

2.3.13. Assume that

y

fc + 1

/cGN.

lim an — a. Find

lim —= ai + —7= -f -7= + ... 4- -7= n->oo ^ y V2 V3 Vn

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Problems. 2: Sequences of Real Numbers

38

2.3.14. Prove that if {an} is a sequence for which lim (a n +i - an) = a,

n—>oo

then lim — = a.

n—>oo 71

2.3.15. Let lim an = a. Determine n—>oo

an_i

(an

r

ai

\

2.3.16. Suppose that lim an — a. Find n—>oo ,

v

v

f

a

n

t

a

«1

n-l

\

2.3.17. Let k be an arbitrarily fixed integer greater than 1. Calculate

n->oo V \n

J

2.3.18. For a positive arithmetic progression {a n }, find l i m n(ai -... »a w )n n^oo ai + ... + a n

2.3.19. Suppose that {an} is such that the sequence {bn} with 6 n = 2an + a n _i, n > 2, converges to b. Study the convergence of 2.3.20. Suppose that {an} is a sequence such that for some real x. Prove that

lim nxan = a

n—>oo

lim n x (ai • aoo .limf^) , \(n!)

Jimf^V. lim - ¥ = , fcGN.

2.3.24. Show that if lim an = a, then 1

lim

1

n

^ afc > — = a.

n-^oo Inn f—' /c fc=i

2.3.25. For a sequence {a n }, consider the sequence {An} of arithmetic means, i.e. An = Q i+ a ?+--+ Q n. Show that if lim^oo j4 n = A, then also

iim J - y ; ^ = A

n->oo I n n k=\ f—' A;

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Problems. 2: Sequences of Real Numbers

40

2.3.26. Prove the converse to the Toeplitz theorem stated in 2.3.1: Let {cn^k : 1 < fc < n, n > 1} be an array of real numbers. If for any convergent sequence {an} the transformed sequence {bn} given by setting n

bn = ^ c n > f c a f c ,

n > 1

fc=i

is convergent to the same limit, then (i)

cn,fc —• 0 for each fceN, n—xx>

n

(ii)

J^Cnt

—• 1,

(iii)

there exists C > 0 such that for all positive integers n n

2.4. Limit Points. Limit Superior and Limit Inferior 2.4.1. Let {a n } be a sequence whose subsequences {a2k}, {&2/c+i} and {03^} are convergent. (a) Prove that the sequence {an} is convergent. (b) Does the convergence of any two of these subsequences imply the convergence of the sequence {a n }? 2.4.2. Does the convergence of every subsequence of {an} of the form {a 5 . n }, s > 1, imply the convergence of the sequence {an}? 2.4.3. Let {a P n }, {aqn},... ,{a S n } be subsequences of {an} such that the sequences {p n }, {qn}, •••» {s n } a r e pairwise disjoint and form the sequence {n}. Show that, if S , S p , S q , ...,S S are the sets of all the limit points of the sequences {a n }, {aPn }, {aqn },..., {a 5n }, respectively, then

s = spusqu...uss.

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2.4. Limit Points. Limit Superior and Limit Inferior

41

Conclude that, if every subsequence {aPn}, {a q n },..., {aSn} converges to a, then the sequence {a n } also converges to a. 2.4.4. Is the above theorem (Problem 2.4.3) true in the case of infinitely many subsequences? 2.4.5. Prove that, if every subsequence {anfc} of a sequence {an} contains a subsequence {ank } converging to a, then the sequence {an} also converges to a. 2.4.6. Determine the set of limit points of the sequence {a n }, where (a)

an=

(b)

an =

(c) (d) (e) (f)

v/4(-D"+2) 1 / [n-l] \ / - 3 - -3 2\ 3 ) ( (l-(-l)")2n + l

[n-l] 3

(1 + cos nn) In 3n + In n /

n-K\n

an = 1 cos — 1 , 2n 2

«n = - ^

[2n2] 7

2.4.7. Find the set of all the limit points of the sequence {an} defined by (a)

an = na-

(b)

a n = na - [na], a 0 Q,

(c)

a n = sin7rna,

a € Q,

(d)

a n = sin irna,

a $ Q.

[na], a G Q,

2.4.8. Let {ak} be a sequence arising by an arbitrary one-to-one indexing of the elements of the matrix { tfn — v^ro}, n, ra E N. Show that every real number is a limit point of this sequence.

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42

Problems. 2: Sequences of Real Numbers

2.4.9. Assume that {an} is a bounded sequence. Prove that the set of its limit points is closed and bounded. 2.4.10. Determine lim an and lim an, where 2n 2

(a)

"2n 2 l

.7 J

71 — 1 U7T —-COS—-,

(b)

0

- °°

2.4.28. Prove that (a)

lim (max{a n ,6 n }) = m a x ] lim a n , lim bn \ ,

(b)

lim (min{a n ,6 n }) = min < lim a n , lim 6 n > . n—>-oo

Ln—>oo

n—>oo

)

Are the equalities (c) (d)

lim (min{a n ,6 n }) = min \ lim a n , lim 6 n I , lim (max{a n ,6 n }) = max < lim a n , lim 6 n > rc—•oo

Ln—>oo

n—>oo

J

also true? 2.4.29. Prove that every sequence of real numbers contains a monotonic subsequence.

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47

2.5. Miscellaneous Problems

2.4.30. Use the result in the foregoing exercise to deduce the BolzanoWeierstrass theorem: Every bounded sequence of real numbers contains a convergent subsequence. 2.4.31. Prove that for every positive sequence {a n }, lim

> 4.

Show that 4 is an optimal estimate.

2.5. Miscellaneous Problems 2.5.1. Show that, if lim an = +oc or lim an = —oo, then n—•oo

n—• oo

lim

n-+oo \

1+ — )

= e.

a„

2.5.2. For x G l , show that

lim (l + -Y = ex. 2.5.3. For x > 0, establish the inequality < \n(x -f 1) < x. v x+2 ' Prove also (applying differentiation) that the left inequality can be strengthened to the following: x

*2iX

x+1


0.

2.5.4. Prove that (a) (b)

lim n{ tya — 1) = In a, a > 0,

n—>-oo

lim n(tfn-l)

= +oo.

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Problems. 2: Sequences of Real Numbers

48

2.5.5. Let {an} be a positive sequence with terms different from 1. Show that if lim an = 1, then 71—>OC

lim

n—»oo a n — 1

2.5.6. Let

= 1.

*» = 1 + I[ + i + "- + ^

Show that

lim a n = e

and

nGR

0 < e — a n < —-.

n—»oo

77,71!

2.5.7. Prove that rp

l i m ( 1 +

l!

T

+

rpli

2 !

+

-

+

^!1=:e*-

2.5.8. Show that

lim ( - + —^- + ... + i - J = In2,

(a) (b) lim ( . *-*°°\y/n{n+l)

+ y/{n+l)(n

+ 2)

+ ... + — = = = = = ) = ln2. ^/2n{2n + 1) /

2.5.9. Find the limit of the sequence {an}, where

2.5.10. Let {an} be the recursive sequence defined by a\ = 1,

a n = n(a n _i + 1)

for

n = 2,3,....

Determine lim

TT ^

+

Mm

2.5.11. Prove that lim (n!e - [n!e]) = 0.

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2.5. Miscellaneous Problems

49

2.5.12. Given positive a and 6, show that

n—•oo \

s^V-v s Z

2.5.13. Let {a n } and {&n} be positive sequences such that lim 0%= a, n—>oo

lim b™ = 6,

where

a, 6 > 0,

n—>oo

and suppose that positive numbers p and g satisfy p + q = 1. Prove that lim (pan + qbn)n = apbq. n—>oo

2.5.14. Given two real numbers a and 6, define the recursive sequence {an} as follows: a\ = a,

a 2 = 6,

an+i =

Find lim an.

n

an + —a n _i, n

n > 2.

n—+oc

2.5.15. Let {an} be the recursive sequence defined by a\ = 1,

a 2 = 2,

a n + i = n(a n -f a n _i),

n > 2.

Find an explicit formula for the general term of the sequence. 2.5.16. Given a and b, define {an} recursively by setting oi = a,

a 2 = 6,

1 2n-l an+i = —an_i H an, 2n 2n

n > 2.

Determine lim a n . n—• oo

2.5.17. Let

°» = 3 - E f c ( i b + 1 j ( f c + 1)I. » €N . (a) Show that lim a n = e. n—>oo

(b) Show also that 0 < an - e
oc

2.5.19. Suppose that {an} is a sequence such that an < n, n = 1,2,..., and lim an = -hoc. Study the convergence of the sequence

( ! - £ ) " . n-1,2,.... 2.5.20. Suppose that a positive sequence {bn} diverges to -foo. Study the convergence of the sequence h \

n

1 + -M , n = l,2,.... nJ 2.5.21. Given the recursive sequence {an} defined by setting 0 < ai < 1,

an+i = an(l - an),

n > 1,

prove that (a)

lim nan = 1,

n—>oo

lim n ( 1 " n a " ) = 1. n-+oo In n

(b)

2.5.22. The sequence {a n } is defined inductively as follows: 0 < a\ < 7T, Prove that

«n+i = sina n ,

n > 1.

lim y/nan — y/3. n—*oc

2.5.23. Let a\ — 1,

a n + i = an + —

,

n > 1.

fc=i

Prove that n

hm ,/ = 1. ^°° v 2hm

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2.5. Miscellaneous Problems

51

2.5.24. For {an} defined inductively by a\ > 0, determine

a n +i = arctana n ,

n > 1,

lim an • n—>oo

2.5.25. Prove that the recursive sequence defined by 0 < ai < 1,

an+\ = cosa n ,

n > 1,

converges to the unique root of the equation x = cos x. 2.5.26. Define the sequence {an} inductively as follows: a\ = 0,

a n + i = 1 — sin(a n — 1),

Find

n > 1.

1 n lim — V a f c .

n—• oo 77, ^ — '

2.5.27. Let {an} be the sequence of consecutive roots of the equation tan a; = x, x > 0. Find lim (a n +i — a n ). n—»oo

2.5.28. For |a| < | defined by

and ai £ M, consider the recursive sequence a

n+i = asinan,

n > 1.

Study the convergence of the sequence. 2.5.29. Given a\ > 0, consider the sequence {an} defined by setting an+i = ln(l + a n ),

n > 1.

Prove that (a) (b

lim nan = 2,

n—»oo

hm n-^oo

Inn

= -. 3

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52

Problems. 2: Sequences of Real Numbers

2.5.30. Define the recursive sequence {an} by putting a\ = 0 and

/l\an an+i = 1 — 1

,

n > 1.

Study the convergence of the sequence. 2.5.31. Given a\ > 0, define the sequence {an} as follows: an+i = 2 ~ a n ,

n > 1.

Study the convergence of the sequence. 2.5.32. Find the limit of the sequence defined by ai = >/2,

an+i=2^\

n > 1.

2.5.33. Prove that if lim (an - a n _2) = 0, then n—>oo

lim

n—•oo

fl

= 0.

2.5.34. Show that if for a positive sequence {an} the limit lim n I 1 - ^ ± ± exists (finite or infinite), then lim — ^ n^oo Inn also exists and both limits are equal. 2.5.35. Given a\,b\ £ (0,1), prove that the sequences {an} and {bn} defined by an+i = a i ( l - a n - bn) + a n ,

&n+i = &i(l - a n - bn) + 6 n ,

n > 1,

converge and find their limits. 2.5.36. For positive a and ai, consider the sequence {an} defined by setting an-K = a n(2 - oa n ), n = 1,2,.... Study the convergence of the sequence.

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2.5. Miscellaneous Problems

53

2.5.37. Show that if a\ and a,2 are positive and «n+2 = \f^a + \J&n+\-> 71 = 1, 2, ..., then the sequence {an} converges. Find its limit. 2.5.38. Assume that / : R+ —> E + is a function increasing with respect to every variable and there exists a > 0 such that f(x,x,...,x)

>x

for

0 < x < a,

f(x,x,...,x)

a.

Given positive oi,a2, ... ,afc, define the recursive sequence {a n } by fln = / ( f l n - l , f l n - 2 , . » j f l n - f c )

f°r

n > A.

Prove that lim an = a. n—*oo

2.5.39. Let a\ and a,2 be given positive numbers. Study the convergence of the sequence {an} defined by the recursive relation a n + i = anean~Q"n-1

for n > 1.

2.5.40. Given a > 1 and x > 0, define {a n } by setting ai = a x , a n +i = a a n , n G N. Study the convergence of the sequence. 2.5.41. Show that Y / 2 + V / 2 + ... + V/2 = 2 c o s - ^ I . N

v

n roots

'

Use this relation to find the limit of the recursive sequence given by setting ai = A/2, a n + i = >/2-fa n , n > 1. 2.5.42. Let {e n } be a sequence whose terms are equal to one of the three values -1, 0, 1. Establish the formula ei]J2 + e2^2 + ... + e„V2 = 2sin (^ ] T

£l

g:;£fc j ,

n G N,

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Problems. 2: Sequences of R e a l N u m b e r s

54

and show that the sequence + £n V2

an = €iy2 + s2y2-{-... converges. 2.5.43. Calculate

,. ( 1 1 1 \ lim arctan - + arctan ———72 -f ... + arctan -—7 n^oc y 2 2•2 2n^ / 2.5.44. Find lim sin(7r\/n2 + n). n—•oo

2.5.45. Study the convergence of the recursive sequence defined as follows: ai

\/2,

a 2 - V 2 + v7^,

an+2 = y

2

+ V3 + a n for n > 1.

2.5.46. Show that

Jirr^ J 1 + 2if 1 + 3 y 1 + ...\]l + {n-\)y/Y+n

= 3.

2.5.47. Given a > 0, define the recursive sequence { a n } by putting a\ < 0,

a n +i =

1 for n € N. an Show that the sequence converges to the negative root of the equation x2 -f x = a. 2.5.48. Given a > 0, define the recursive sequence {an}

by setting

a an+i = for n G N. an + 1 Show that the sequence converges to the positive root of the equation x2 4* x — a. Q>\ > 0,

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2.5. Miscellaneous Problems

55

2.5.49. Let {an} be the sequence defined by the recursive formula 2 + an tor n G N. 1 + an Show that the sequence is Cauchy and find its limit. ai = 1,

an+i =

2.5.50. Show that the sequence defined by ai>0,

an+i=2H

, n € N,

is Cauchy and find its limit. 2.5.51. Given a > 0, define {a n } as follows: ai = 0,

an+i =

2 4-o n Study the convergence of the sequence.

for n e N.

2.5.52. Assume that aiGl

a n + i = \an - 2 1 _ n |

and

for n 1, then

,.

n^V~l

1

lim — > —— = i 7n-^oo bn ^—' 7 6-1

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Problems. 2: Sequences of Real Numbers

56 2.5.54. Calculate lim

sin

n—•oo

n+ 1

-f sin

n+ 2

+ ... + s m — . 2nJ

2.5.55. Find lim TT [ 1 + —«4 1 , n-+oo -LX V en* 7/ x

(a)

where

c > 0,

k=i

A / lim TT ( 1

(b)

n-^oo

± ±

k=l

\

A:2 \ =• J ,

cn6

J

where

c > 1.

2.5.56. Determine lim n-+oo

— TT sin — = . n\ x± nJn k~\

2.5.57. For the sequence {an} defined by

a =

n

y

v

-1

- S (fcj '

show that

n

^ X'

lim a n = 2. n—•oo

2.5.58. Determine for which values of a the sequence

converges. 2.5.59. For x G R, define {x} = x - [x]. Find lim {(2 + \/3) n }n—>oo

2.5.60. Let {a n } be a positive sequence and let Sn = a\ -\-a,2 +... + a n , n > 1. Suppose that
n+l

((5 n - l)an + a n _ i ) , n > 1.

Determine lim a n .

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2.5. Miscellaneous Problems

57

2.5.61. Let {an} be a positive sequence such that i-

an

Find

a

r~

A

hm — = 0,

hm

n—»oo 77,

l +a2

n—>oo

lim

+ — + an ft

^

< oo.

aj + al+ ... + oo a\ 4- a 0 there is a positive integer no such that |flfc,n _

I bk,n

1

I

< e

for each n > UQ and k = 1,2,..., n. Show that if lim ^ 6/~ n exists, n

then

lim V V

n

~'OG

k=i

= lim V V n . fc=l

fc=l

2.5.83. Given a ^ 0, find n

lim V s i n k=l

(2fc-l)a n^

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2.5. Miscellaneous Problems

61

2.5.84. For a > 0, determine

2.5.85. Find

lim f[(l fc=l

+JL). x

7

2.5.86. For p y^O and g > 0, determine

2.5.87. Given positive numbers a, b and d such that b > a, calculate a(a + d)...(a + nd) n^oo 6(6 + d)...(6 + nd) '

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Chapter 3

http://dx.doi.org/10.1090/stml/004/03

Series of Real Numbers

3.1. Summation of Series 3.1.1. Find the infinite series and its sum if the sequence {Sn} of its partial sums is given by setting (a) (c)

Sn = ^—, n €N, n Sn = aretann, n G N,

(b)

Sn = — - i , n €N, 2n n f-l) 5 n = -——, n G N. n

(d)

3.1.2. Find the sum of the series

^ n£ (=n +&l ) '• 2

n=l oo

(C)

2

v

' / o 2

w

w^ (£2 n - l ) ( 2 n + l) ' n—\ oo

r

v-^ n - Vn - 1

S^TTT

(e) ~£ (v^+\/n + 1)^71(71+1)

2

2

'

v

'

1

1

S=^.

63

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Problems. 3: Series of Real Numbers

64

3.1.3. Compute the following sums: X

i

( ^ a

^Sr^

m

fo+l)(3n+l)

()

m

(b)

Sln(n+i)(2n-l)'

7 + /

n=l

77T1 + 4) T:—~ > n(3n

"

(2n + l)n

3.1.4. Find the sum of the series

V

(a)

^

L

n ( n + ll)...(n ) . . . + ra) v

n=l oo

, m 6 N,

'

1

r, m e N , *—' nln-f m)

(b)

V - T — n=l

'

E- J (n + l)(n + 2)(n + 3)(n + 4)'

(c)

3.1.5. Compute OO

,

OO

w S>£.

w E;

n=l

3.1.6. Calculate

OO

E

n=l

1

n=l

Inn I n — In n

j

sin —2^+1

COS

2n+1'

3.1.7. Find

E n!(n -h1n 4

n=0

2

+ l)'

3.1.8. Show that OO

E. 3 - 5 - . . . - ( 2 n + l )

n=l

_,

n

1

=

2"

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3.1. Summation of Series

65

3.1.9. Suppose that {an} is a sequence satisfying lim ((en + l)(a 2 + l)...(a n + 1)) = p,

0 < g < +oo.

n—>oo

Prove that oo

_ 1_ 1

On

n=1(ai

+ l)(a 2 + l)...(an + l) "

p'

where we assume that — = 0. oo

3.1.10. Using the result in the foregoing problem, find the sum of the series Y^ n — 1

(a)

n— 1

y, 2n-l ^ 2 - 4 - 6 - . . . -2n'

(b)

n=l oo

(c)

i

S(i-^)(i-l)-(i-^)'

3.1.11. Let {an} be a recursive sequence given by setting a\ > 2, Show that

y>

a n + i = a^ - 2 for

n € N.

_ ai - v/a? - 4

1

^-J ai • a 2 •... • an

2

n=l

3.1.12. For b > 2, verify that

E

71!

^b(6+l)...(b +

ra-l)

6-2'

3.1.13. For a > 0 and 6 > a -f 1, establish the equality

E

a(a -f l)...(a -f n — 1) 6(6+l)...(6 + n - l )

=

a 6-a-l'

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Problems. 3: Series of Real Numbers

66

3.1.14. For a > 0 and b > a -f 2, verify the following claim: °°

a(a + l)...(a + n - 1) _

E fc(6+l).»(6 + n - l )

n==i

n

a(6 - 1)

~ (6-a-l)(6-a-2)'

3.1.15. Let J^ — be a divergent series with positive terms. For 71=1

aU

b > 0, find the sum

£~ ( 2 + &)( 3 + b)...(On+l + fc) ai • a,2 •... • an

a

a

3.1.16. Compute

f(-ir^.

n=0

3.1.17. Given nonzero constants a, 6 and c, suppose that the functions / and g satisfy the condition f(x) = af(bx) -f cg(x). (a) Show that, if lim anf(bnx)

= L(x) exists, then

n—•oo

oo

Y,ang{bnx)

=

f(x) - L(x)

n=0

(b) Show that, if Jim a~nf(b~nx)

= M(x) exists, then

n—+oo oo

^a-ng{b-nx)

=

M{x) -

af{bx)

n=0

3.1.18. Applying the identity sinx = 3 sin | - 4 sin 3 | , show that 3

x-sini

x

n=0 oo

i1

n

n=0

3n

sin 3

x

3 . x

-r = - sin —. 3~n+1 4 3

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67

3.1. Summation of Series

3.1.19. Applying the identity cot a; = 2cot(2x) + tan a; for x ^ /c|, fcGZ, show that

E — tan — = n=0

l

X

1

2n

2n

x

3.1.20. Using the identity establish the formulas

^

/^ \

2 cot (2a;).

arctan x = arctan(frr) + arctan

1+bJ?,

n

\~^

(a) > arctan

(11 ~- ub)b K )u

xx ————^ = arctanx

n=0

for

0 < b < 1,

for

x ^ 0

n

v~> (b — l)b x (b) >^ arctan , 0 ,. 0 = arccot x w ^ 1 + '62 n + 1 x 2 n=0

and

6 > 1.

3.1.21. Let {a n } be the Fibonacci sequence defined by setting a 0 = a\ — 1, and put 5 n =

a n + i = a n + a n _i, n > 1,

fc=0

v (-ir ^

Sn

3.1.22. For the Fibonacci sequence {an} defined in the foregoing problem, calculate

y

(-l)n

3.1.23. For the Fibonacci sequence {an} defined in 3.1.21, determine the sum of the series oo

V^ arctan

1

.

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Problems. 3: Series of Real Numbers

68 3.1.24. Find the sum: oo 2 y ^ arctan — ,

(a)

°° 1 V^ arctan —x

(b)

n=l oo

(c)

n=\

> arctan n=l

,

n 4 - 2 n 2 + 5*

3.1.25. Let {an} be a positive sequence that diverges to infinity. Show that oo

E arctan

n

— Q>n

an4-l

-L

= arctan —. 1 + anan+i ai 3.1.26. Prove that any rearrangement of the terms of an infinite positive series does not change its sum. n=1

3.1.27. Establish the identity oo

oo O ^r—'\

1

E (2n-l)

0

l

2 =

1

1

4^n?'

3.1.28. Prove that °° 1

(a)

£^

n=l oo

(b)

^ ^

.

=

7T2

4

= ^0'

oo

¥' .

B-«"5r+T " I-

n=0

3.1.29. For the sequence {a n } defined recursively by a\ = 2,

a n + i = an-

an + l

for n > 1,

oo

n=l

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69

3.1. Summation of Series 3.1.30. Let {an} be defined as follows: a\ > 0,

a n + i = In

ean — 1

for n > 1,

and set bn = a\ • a^ •... • an. Find ]T 6 n . n=l

3.1.31. Let {a n } be defined by setting ai = 1,

an+i =

\/2

ai + a 2 -f ••• 4- a n

for

n > 1.

oo

Determine the sum of the series ^ a n . n=l

3.1.32. Find the sum of the following series:

(a)

E(-l)-1^ n=l

n(n + l ) '

™=i

°° /

1

1

1

\

3.1.33. Calculate

D-i)"-b.(i + i).

n=l

x

'

3.1.34. Compute

3.1.35. Determine the sum of the series

S(=-H))-

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Problems. 3: Series of Real Numbers

70

3.1.36. Suppose that a function / is differentiable on (0, oo), that its derivative / ' is monotonic on a subinterval (a, H-oc), and lim fix) = x—++oo

0. Prove that the limit lirn^ Q / ( l ) + /(2) + /(3) + ... + f(n - 1) + ± / ( n ) - £

f(x)dx^J

exists. Consider also the special cases of the functions f(x) = ^ and f(x) = lnx. 3.1.37. Determine the sum of the series ,lnn

E(-D- n

71=1

3.1.38. Find

f / , 2 n + l > ^

n In V x

n=l

2n-l

3.1.39. Given an integer k > 2, show that the series y>/ 1 2s ^ \[((n n --l )1f )k c +4-1 l

+

1 1 + + (n - 1 )k+42 9 """" n * - l (ra-l)A;

nA:

converges for only one value of x. Find this value and the sum of the series. 3.1.40. For the sequence {an} defined by setting a 0 = 2,

a n + i = an H

3 + (-l)n

,

compute n=0

3.1.41. Prove that the sum of the series oo n=l

-

oc n=l

1

v

'

is irrational.

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3.1. Summation of Series

71

3.1.42. Let {en} be a sequence where en is either 1 or — 1. Show oo

that the sum of the series Y ^

is an irrational number.

U

n=l

'

3.1.43. Show that for any positive integer k the sum of the series (-i)n (n\)k

" ^

v

n=l

'

is irrational. 3.1.44. Suppose that {n^} is a monotonically increasing sequence of positive integers such that hm

fc^oo niTl2 • ... • Tlk—l

= -foo.

oo

Prove that Y] — is irrational. 2=1

3.1.45. Prove that, if {n^} is a sequence of positive integers such that lim = +oo /c-+oo mn2 • ... • njfe_i

and

lim ^ ^

> 1, nk-\

oo

then V — is irrational. n .—' i 1= 1

3.1.46. Suppose that {n^} is a monotonically increasing sequence oo

of positive integers for which lim ^v/n^ = oo. Prove that J ] ^- is k-^oc

k=1

k

irrational. CO

3.1.47. Let Y" 2^, where pn,Qn £ N, be a convergent series and let —- w n=l

Pn

qn - 1

Pn+1

qn+i - 1

Pn

qn

Denote by A the set of all n for which the above inequality is sharp. oo

Prove that Yl ~

1S

irrational if and only if the set A is infinite.

n=l

3.1.48. Prove that for any strictly increasing sequence {n^} ofposoo

itive integers the sum of the series ^T ^-j is irrational.

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Problems. 3: Series of Real Numbers

72

3.2. Series of Nonnegative Terms 3.2.1. Determine whether the following series converge or diverge: oo

oo

(a) £ ( v ^ T T - ^ T T ) ,

n=l \

^(2n-3)!!

...

(d)

w £(2^jM' n=2 00

(e)

£ n=l

v

'

/ V

i \

1-cos-

n /

,

'

A /

n \

\

n2

•

' / n(n+l)

^UTTJ N

n=l °°

(f)

'

£(^-l)», n=l

OO

(g)

2 i i

(b) E ( ^ ~ ^ T

n=l

.

•

^(^S-l), a>l. n=l

3.2.2. Test the following series for convergence:

(a)

g l l n f l + iV v

n=l

W

(b) g 1 ln=±l,

'

n=2

(d)

Ln2_lnn'

n=l OO

(e)

v

2^(l n n )lnn

n=2

v

y

-.

2 ^ n=2

H n n V1 n l n n " (Inn)

3.2.3. Let Y2 ani Yl bn be series of positive terms satisfying n=l

n=l &n+l

. ^n+1

^

r

< —— for n > n 0 .

oo

oo

Prove that if Yl bn converges, then Y2 an ^so converges. n=l

n~l

3.2.4. Test these series for convergence: (a)

D^T' n=l

(b)

ZLT^An=l

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3.2. Series of Nonnegative Terms

73

3.2.5. Determine for which values of a the given series converges.

(a)

oo

oo

(b) £ ( ^ - l ) a ,

£ ( ^ - 1 ) * , a>l,

n=l 00

1\

//

n+1

( E f ( i + s ) n=l

N

\

n=l

\

a

i \a

°° /

-«) - «) £ ( * — - J x

n=l

/

'

/

oo

3.2.6. Prove that, if a series ^ an with positive terms converges, n=l

then

OO

y ^ (a a n - 1),

where

a > 1,

n=l

also converges. 3.2.7. Investigate the behavior (convergence or divergence) of the following series: /

OO

1\

°°

(a)

^-lnfcos-j, n=l V n^

(c)

/ ^

y % **""+*, n=l

a,6,c,deR,

n2n

°^

n=l

(b)

T x—TT7 r^—r~ > a, 6 > 0. (n -h a) n + fc (n + b)n+a v

/

\

/

oo

3.2.8. Suppose a series ^ a n of nonnegative terms converges. Prove 71=1 OO

that ]P y/anan+i

also converges. Show that the converse is not true.

71=1

If, however, the sequence {an} is monotonically decreasing, then the converse statement does hold. oo

3.2.9. Assume that a positive-term series ^ an diverges. Study the behavior of the following series:

(a)

w

oo

Sift' ^irfe'

71=1 OO

n=l

(b) (d)

71=1

oo

^rrk' Ei^f71=1 OO

n=l

n

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Problems. 3: Series of Real Numbers

74

3.2.10. Assume that a positive term series ^ an diverges. Denote n=l

by {Sn} the sequence of its partial sums. Prove that oo

—

n=l

and

oo

diverges

—

converges.

n=lSn

3.2.11. Show that under the assumptions of the preceding proposition the series oo

E converges for each (3 > 0.

a

n

q qP

n=2 °n>->n-l

3.2.12. Prove that under the assumptions of 3.2.10 the series oo

E

n=l

a

n n

converges if a > 1 and diverges if a < 1. oo

3.2.13. Prove that, if a positive term series 7^2 converges and 1 = 1 an rn =

Yl ft/e, n G N, denotes the sequence of its remainders, then

k=n+l

oo

(a)

V^ ——

diverges,

n=2r"-l

converges. 3.2.14. Prove that under the assumptions of the foregoing problem oo

converges if a < 1 and diverges if a > 1.

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75

3.2. Series of Nonnegative Terms

3.2.15. Show that under the assumptions of Problem 3.2.13 the series 53 a n+i In rn converges.

n=l

oc

3.2.16. Let Yl an be a series of positive terms. Suppose that n=l

lim n In —— = #.

n->oo

Prove that

an+i

oo

53 an converges if g > 1 and diverges if g < 1 (the

n=l

cases # = +oo and # = — oo are included). Show that if g = 1, then the test is inconclusive. 3.2.17. Study the behavior of the following series: oo

oo

1

n=l

(d)

00

oo

1

n=l -.

E ^

a>0

n—1

..

n=l 00

..

(e) E ^ n ^ '

>

n=2

a>0

-

3.2.18. Discuss convergence of the series 00

£V + * + -+*,

a >0.

n=l

3.2.19. Use the result of Problem 3.2.16 to prove the limit form of the Test of Raabe. Let an > 0, n G N, and let lim n [ —

n->oo

\a

n +

i

1 1 — r.

00

Prove that 53 a™ converges if r > 1 and diverges if r < 1. n=l

3.2.20. Let {a n } be defined recursively by setting Q>1 = «2 = 1,

« n + l = &n H

ofln-l

f°r

™ > 2.

oo

Study convergence of the series 53 ~ • 71=1

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Problems. 3: Series of Real Numbers

76

3.2.21. Let a\ and a be positive. Define the recursive sequence {an} by putting a n + i = ane~a"

for

n=l,2,.... oo

Determine for which values of a and f3 the series J2 an converges. n=l

3.2.22. Determine for which values of a the series n! ^ ( a + l ) (( ao - + 2 ) . . . . - ( a + n) converges. 3.2.23. Let a be an arbitrary positive number and let {bn} be a positive sequence converging to b. Study the convergence of the series

E

n=1

nlar (a + 6i)(2a 4-fe2)• - • (na + 6 n )'

3.2.24. Prove that, if a sequence {a n } of positive numbers satisfies Qn+l __ i _ 1

7n

n

ttrj

n In n'

OO

where 7 n > T > 1, then ^ a n converges. On the other hand, if 71=1

Qn+1

^n

i _ 2.

OTI

n

n In n '

oo

where 7 n < T < 1, then ^ a n diverges. (This is the so-called Test of Bertrand.)

71=1

3.2.25. Use the tests of Bertrand and Raabe to derive the following Criterion of Gauss. If {an} is a sequence of positive numbers satisfying Qn+1 _ ,

an

PL

n

An

nx' oo

where A > 1 and {$ n } is a bounded sequence, then J2 an converges when a > 1 and diverges when a < 1.

n=l

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3.2. Series of Nonnegative Terms

77

3.2.26. Discuss the convergence of the series y , a(a + 1) • ... • (a + n - 1) /?(/? + 1) •••••(/? + n - 1) ^ n! 7(7 + 1) •... . ( 7 + n - 1)' where a,/3 and 7 are positive constants. 3.2.27. Determine for which values of p '(2n-l)!!\p ^-f V (2n)!! converges. 3.2.28. Prove the following condensation test of Cauchy. Let {a n } be a monotonically decreasing sequence of nonnegative 00

numbers. Prove that the series Yl an converges if and only if the n=l

00

series ]T 2 n a2^ converges. n=l

3.2.29. Test the following series for convergence: 00

00

1

^n(mn)a'

n=2

v

'

'

^

n=3

-

n • Inn • l n l n n '

3.2.30. Prove the following Theorem of Schlomilch (a generalization of the Cauchy theorem, see Problem 3.2.28). If {gk} is a strictly increasing sequence of positive integers such that for some c > 0 and for all k 6 N, gk+i — gk < c(gk — 9k~i) and if a positive sequence {an} strictly decreases, then ^

a n < 00

if and only if

^ ( # f c + i - gk)a9k < 00.

3.2.31. Let {a n } be a monotonically decreasing sequence of positive numbers. Prove that the series Yl an converges if and only if the following series converges:

71=1

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Problems. 3: Series of Real Numbers

78

(a)

^3na3n,

(b)

n=l

^nan2,

(c)

n=l

^n2an3. n=l

(d) Apply the above tests to study convergence of the series in Problem 3.2.17. 3.2.32. Suppose that {an} is a positive sequence. Prove that lim (an)^^ n—>oo oo

< C

implies the convergence of J2 an. 71=1

3.2.33. Suppose that {an} is a positive sequence. Show that lim (na™)^ 1 ^ < n—->oo oo

C

implies the convergence of ^2 an. 71=1

3.2.34. Let {a n } be a monotonically decreasing sequence of positive numbers such that 2 CiOn

— OC

OO

3.2.37. Let J2 an be a convergent positive series. Give necessary 71=1

and sufficient conditions for the existence of a positive sequence {bn}

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3.2. Series of Nonnegative Terms

79

such that the series oo

oo

and

6

Yl "

Yl T

n=l

n—1

both converge. 3.2.38. Does there exist a positive sequence {an} such that the series OO

OO

Y] an ^—

and

n=\

both converge. 3.2.39. Show that

oo

-.

V " —— ^ n2an l

n=l

1

V^ _

1

"* Q n + 1

n=l

diverges for any positive sequence

{an}.

3.2.40. Let {an} and {bn} be monotonically decreasing to zero and oo

oo

such that the series ^2 an and J2 bn diverge. What can be said n=l

n=l

oo

about the convergence of ]T) cn, where cn — min{a n ,6 n }? n=l

3.2.41. Let {an} be a monotonically decreasing sequence of nonnegoo

ative numbers such that Y2 ^f diverges. Assume that n=l

bn = min 0 OO

„=i

a

converges. ^
oc

3.2.48. Let {an} be a positive sequence diverging to infinity. What can be said about the convergence of the following series: oo

1

(a) y±,

n=l ""

oo oo

^

oo

(b) Y4-, n=l

^

(C) y-,4-'

I

-—.

n=l " n

"«

I

_? In n

3.2.49. Study convergence of ^ a n , where n=l

ai = 1,

a n + i = cosa n

for

nG

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3.2. Series of Nonnegative Terms

81

3.2.50. Let p be an arbitrarily fixed nonnegative number. Study the OO

convergence of Y2 a n , where n=l

an+i = n_psinan

ai = 1,

for

n G N.

3.2.51. Let {a n } be a sequence of consecutive positive solutions of oo

the equation tanx = x. Study the convergence of Y2 ^-. n=l

an

3.2.52. Let {a n } be a sequence of consecutive positive solutions of oo

the equation tan y/x — x. Study the convergence of Yl ~ • n=l ° n

3.2.53. Let a\ be an arbitrary positive number and let a n +i = oo

ln(l + an) for n > 1. Study the convergence of Yl an71=1

3.2.54. Assume that {an} oo

is a positive monotonically decreasing

a

sequence such that Y2 n diverges. Show that 71=1

Ql + Q 3 + ... +a2n-l

y rwoo

_ -

a 2 4" a4 + ... + &2n

3.2.55. Let 5^ = 1 + ^ + ... + ^ and let kn denote the least of all positive integers k for which Sk >n. Find ,. &n+i lim — — . n—>oo kn

3.2.56. Let A be the set of all positive integers such that their decimal representations do not contain zero. (a) Show that

^2 n neA

conver

ges.

(b) Determine all a such that Y2 ^ n€A

converges.

oo

3.2.57. Let Y2 an be a series of positive terms and let 71=1

lim —2=- = p. n-^oo Inn

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Problems. 3: Series of Real Numbers

82

Prove that if g > 1, then the series converges, and if g < 1, then the series diverges (here g may be equal to +oo or — oo.) Give examples showing that in the case of g = 1 the criterion is indecisive. 3.2.58. Show that the Raabe test (see Problem 3.2.19) and the test given in Problem 3.2.16 are equivalent. Moreover, show that the criterion in the preceding problem is stronger than each of the above mentioned tests. oo

3.2.59. Study the convergence of ^ an whose terms are given by 71=1

Ol

Vi,

2 - y 2 + V 2 + ... + y/2,

N

n>2.

(n — 1)—roots

3.2.60. Let {an} be a sequence monotonically decreasing to zero. Show that if the sequence with terms (oi - an) + (a 2 - an) + ... + (a n _i - an) oo

is bounded, then Yl an must converge. n=l

3.2.61. Find a series whose terms an satisfy the following conditions: fll = 2 '

«n = « n + l + « n + 2 + •••

for

72 =

1,2,3,.... oo

3.2.62. Suppose that the terms of a convergent series Yl an whose n=l

sum is S satisfy two conditions: ^i > a2 > a 3 > ...

and

0 < an < an+\ + a n + 2 + ...,

n £ N.

Show that it is possible to represent any number 5 in the half-closed 00

interval (0, S] by a finite sum of terms of the series Yl an 00

infinite subseries Y2 ank, where {ank}

n=l

is a subsequence of

or

by an

{an}.

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83

3.2. Series of Nonnegative Terms

3.2.63. Assume that ]T) an is a series of positive and monotonically 71=1

decreasing terms. Prove that if each number in (0,5), where S denotes the sum of the series, can be represented by a finite sum of oo

terms of {an} or by an infinite subseries J2 anki where {ank} fc=i

is a

subsequence of {a n }, then the following inequality holds: &n < a n + i + «n+2 + ••• for each

n G N.

oo

3.2.64. Let Y2 an be a divergent series of positive terms and let n=l

lim |p- = 0, where S n = a\ + a2 + ... + an. Prove that ..

lim

n-^oo

a i 5 f 1 + a 2 5 ^ 1 + ... + a n 5 - 1 *

^—

.

— — 1.

ln6n

3.2.65. Using the preceding problem, show that n-^oo

Inn

3.2.66. Let Y2 an be a convergent series of positive terms. What n=l

can be said about the convergence of Ql + a 2 + - + Q n 9 n=l

3.2.67. Prove that if {an} is a positive sequence such that ^ ^ a^ > n

2n

oo

fc=n+l

n=l

fc=i

]T} afc for n € N, then ]T a n < 2eai.

3.2.68. Prove the following Inequality of Carleman: If {an} is a positive sequence, then

n=l

n=l

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Problems. 3: Series of Real Numbers

84

provided that ^ an converges. n=l

3.2.69. Show that if {an} positive integer A;,

is a positive sequence, then, for every

oo

oo

•

n=l

N

..

n=l

, i \ n '

3.2.70. Let {an} be a sequence of positive numbers. Prove that the convergence of J2 ^~ implies the convergence of 71=1

n

oo

/

]CI

n2an

• n

(]C x

n=l \

ak

-2^

x

fc=l

3.2.71. Let {an} be a monotonically increasing sequence of positive oo

numbers such that ]T ^- diverges. Show that an

n=l

y

-

oo

^

1

n a n - (n - l ) a n _ i

is also divergent. 3.2.72. Let {pn} be a sequence of all consecutive prime numbers. oo

Study convergence of ^

n=l

—.

3.2.73. Study convergence of oo

.,

I

,

{n-

l)pn-i'

y

~2npn-

where pn denotes the nth prime number. 3.2.74. Evaluate oo

lim

-^

'

k=2

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3.2. Series of Nonnegative Terms

85

3.2.75. Let {an} be a sequence satisfying the following conditions: 0 < an < 1 for all

and

n e N

a\ ^ 0.

Let Sn = cti + ... -f an

and

Tn = Si + ... + Sn. oo

Determine for which values a > 0 the series £3 ^ - converges. n

n=l

3.2.76. Let A: be an arbitrary positive integer. Assume that {an} is a monotonically increasing sequence of positive numbers such that ^2 -^- converges. Prove that the series

n=l

E

m an

, and

n=l

^-^ In n > n=l

are either both convergent or both divergent. 3.2.77. Assume that / : N —• (0, oo) is a decreasing function and (f : N —» N is an increasing function such that (f(n) > n for all n € N. Verify the following inequalities: ¥>(1)-1

ip(n)-l

n-1

fc=i fc=i fc=i

(2)

V>(n)

£

n

/(*)>E^w)^(*)-v(*-1))-

fc=yj(l)-l fc=2

3.2.78. Prove that under the assumptions of the foregoing problem, if there exists q such that for all n e N the inequality /(y(n))(y>(n+l)-y>(n))

7W

oc

holds, then ]T f(n) n=l

,

-goc an

= g< Z

and divergent when hm

=g>

-.

3.2.80. Derive from Problem 3.2.78 the following test for convergence and divergence of positive series (compare with Problem 3.2.34). A positive series ]P an whose terms are monotonically decreasn=l

ing is convergent provided that hm n—+oc an and divergent provided that lim ^

= g 2.

3.2.81. Using Problem 3.2.77, prove the criteria given in 3.2.31. 3.2.82. Prove the following Test of Kummer, Let {an} be a positive-valued sequence. (1) If there are a sequence {bn} of positive numbers and a positive constant c such that bn—

Gn+l

bn+i > c for all

n £ N,

oo

then J2 an converges. n=l

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3.2. Series of Nonnegative Terms

87

(2) If there is a positive sequence {bn} such that £] ^- diverges n=l

and ^-^--671+1

0 1. Verify the following claims: oo

(a)

^£Sk

diverges

'

oo

(b)

Y^ n

^

converges.

_ 1 ^n l n &n

3.3.7. Let / be a positive and decreasing function on [1, oo). Assume that a function p is strictly increasing, differentiable and such that ip{x) > x for x > 1. Prove that, if there exists q < 1 such that ^

— # ^ or sufficiently large x, then

f(x)

Prove also that, if ^ series diverges.

/(l)

X

^ /(n)

n=l

converges.

- 1 f° r sufficiently large x, then the

3.3.8. Let / , g be positive continuously differentiable functions on (0, oo). Moreover, suppose that / is decreasing. (a) Show that, if lim (-g(x)^verges.

x—•oo ^

- g'(x)\

> 0, then £ fin) n=l

n

(b) Show that, if the sequence with terms J -h\dx l

and for sufficiently large x, —g(x)j^j—gf(x)

con

"

is unbounded

g{x)

< 0, then ]T / ( n ) n=l

diverges.

3.3.9. Let / be a positive continuously differentiable function on (0, oo). Prove that (a) if

lim (-^jffi)

x—>oo ^

f'( )

'

> 1, then £ f(n) converges, n=l

°°

(b) if - Xf(x) < 1 for sufficiently large x, then ^ /(n) diverges. n=l

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Problems. 3: Series of Real Numbers

90

3.3.10. Let / be a positive continuously differentiate function on (0, oo). Prove that (a) if lim X—>00

- ^ J x l n x > 1, then J2 f(n)

[-JT^T ^

converges,

71=1

'

(b) if ( — yr^Y — ~: J x l n x < 1 for sufficiently large x, then Y ^

diverges.

n=l

'

f(n)

3.3.11. Prove the following converse of the theorem stated in 3.3.8. Let / be a positive decreasing and continuously differentiate function on (0, oo). oo

(a) If Yl / ( n ) converges, then there exists a positive continuously 71=1

differentiate function g on (0, oo) such that

^(-• f t OO

(b) If ^ f(n)

diverges, then there exists a positive continuously

71=1

different iable function g on (0, oo) such that the sequence with terms

r I

/h 9{x)-dx, n = 1,2,..., is unbounded and for sufficiently large x,

-«*)£$-•(.>a^0'

V^

ft

n=l

3.4.2. For c 6 l , study convergence and absolute convergence of the series 00

nn-\

n l

na ~

-f Inn

where na is an index depending on a such that nan n > n0.

1

-f In n 7^ 0 for

00

3.4.3. Suppose that a series ^ a n with nonzero terms converges. 71=1

Study the convergence of the series

n=l

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3.4. Absolute Convergence. Theorem of Leibniz

93

3.4.4. Does the condition lim ^ n = 1 imply that the convergence oo

n^oo °

oc

a

of ^2 n is equivalent to the convergence of ^2 bn? n=l

n=l

oo

3.4.5. Assume that a series ]T) an converges conditionally and set n=l

pn =

|an|

2

Qn

, qn =

|Qn|

2

° n . Show that both

£ Pn and X) g n n=l

diverge.

n=l

oo

3.4.6. Assume that a series ]T] a n converges conditionally.

Let

n=l

oo

{P n } and {Q n } be the sequences of partial sums of Yl Pn and n=l

oo

^