Sequences and Series -
J A Green
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Sequences and Series -
J A Green
c c
-. -. .CII CIS
.~
E
CIS G) E"tJ G)
G)
~..J
CIS G) _~:: CIS
o~ ..CIS ....0 ~
.a .._ "tJ ..JW
SEQUENCES AND SERIES BY
J.
LONDON:
A. GREEN
Routledge & Kegan Paul Ltd
NEW YORK:
Dover Publications Inc
First PUblished I958 in Great Britain by Routledge 0- Kegan Paul Limited Broadway House, 68-74 Carter Lane London, E.G.4 and in the U.S.A. by Dover Publications Inc. I80 Vanck Street New York, IOOI4
© J. A. Green I958 Reprinted I959, I962 , I964, I966 No part of this book may be reproduced in any f01'm without permission from the publisher, except f01' the quotation of brief passages in criticism. Library of Congress Catalog Card Number: 66-2I238
Printed in Great Britain by Butler 0- Tanner Limited Frome and London
Preface THIS book is intended primarily for students of science and engineering. Its aims are, first, to present the fundamental mathematical ideas which underlie the notion of a convergent series, and secondly to develop, as far as the small space allows, a body of technique and a familiarity with particular examples sufficient to make the reader feel at home with such applications of infinite series as he is likely to meet in his scientific studies. I do not believe that these two aims are mutually antagonistic. It is true that a certain sophisticated skill is necessary for the construction of proofs of even quite elementary theorems involving, for example, the definition of the limit of a sequence, and that the acquisition of such skill would take more time than the non-specialist mathematician can spare. But this does not mean that either the fundamental definitions or the statements of the theorems cannot be clearly understood by the non-specialist; on the contrary, it is essential that they should be understood. Accordingly it has been my policy to lay more emphasis on the illustration of basic ideas by numerical examples, than on formal proofs; the latter have often been relegated to small print, or omitted (such omissions are noted in the text). In particular the idea of convergence itself is directly involved in the practical problem of numerical calculation of the sum of a series, and I have devoted some space to this topic, traditionally neglected in elementary books on series. It is a great pleasure to acknowledge my debt to my colleagues at Manchester, and especially to Dr. W. Ledermann, for their constructive comments at every stage.
J. The University, Manchester v
A. GREEN
Contents PAGE
Preface
V
CHAPTER I.
Sequences
:r
I.
Infinite sequences
I
2.
Sucressive approximations
2
3. Graphical representation of a sequence 4. The limit of a sequence
3 5
5. Other types of sequence
8
6. Rules for calculating limits
9
7. Some dangerous expressions
12
8. Subsequences 9. Monotone sequences and bounded sequences 10. The functions x",n' and n'x" II.
2.
13 :r5
17
Solution of equations by iteration
20
Exercises
22
Infinite series I.
Finite series
2.
Infinite series
3. Convergent and divergent series 4. Some examples of infinite series 5. Some rules for convergent series 6. A test for divergence 7. The comparison test vii
24 24 25 27 28 30 32 33
CONTENTS PAGE
CHAPTER
8. The ratio test
38
9. The integral test
39
Series with positive and negative terms. Leibniz's test
43
II.
Absolute convergence
45
12.
Power series
47
10.
13. Multiplication of series
51
14. Notes on the use of the convergence tests Exercises
54 55
3. Further techniques and results
58
I.
Numerical calculation of the sum of a series
58
2.
Estimating the remainder of a power senes
61
3. Integration of power series
64
4. Differentiation of power series
70
5. Cauchy's convergence principle
72
6. Dirichlet's convergence test
73
Exercises
75
Answers to exercises
77
Index
78
viii
CHAPTER ONE
Sequences 1. INFINITE SEQUENCES
A sequence is any succession of numbers al , a2, as, •.• ; these numbers are called the terms of the sequence. A finite sequence al , a 2 , ••• , aN is one which has only a finite number of terms, but we shall be interested mainly in infinite sequences whose characteristic property is that they have no last term. For example, the sequence I, 2, 3, ..• of the positive integers is an infinite sequence; so is the sequence I, -t, t, -1, ... whose nth term is (-I)"+1/n. If we want to describe an infinite sequence, it is obviously impossible to write down all its terms. Instead we must give, as in the last example, a rule for calculating the nth term a". This rule for the 'general term' may take the simple form an=f(n), wheref(n) is some easily evaluated function of n; in the two sequences just mentioned, for instance, we had an =n, and a,,=(-I)n+1/n , respectively. It is often useful to write (an) as an abbreviation for the sequence al, ai' as, ••. whose nth term is an' Example 1. Take any fixed number x. Then (x") is the sequence x, X2, xs, .... Example 2. an=n' (s any fixed number) is the general term of the sequence (n')=I', 2', 3', . •• For S=I this is just the sequence (n) of positive integers. Example 3. Take a,,=I, for all n. This defines the sequence (1)=1, I, I, . . . all of whose terms are equal to I. (Notice that it is not necessary that all the terms of a sequence should be distinct.) Example 4. Another example where the terms are not all I
SEQUENCES
distinct is the case %=-1 of Example I. This sequence contains only the numbers I and -I, alternately, viz. -I, I, - I , I, • . .
On the other hand many of the sequences which occur in practice are defined recursively; this means that a rule is given, by which the nth term a.. can be computed when the earlier terms are known. Example 5. If a..+1=V2a.. , we can calculate a..+ t as soon as a.. is known. The value of at must be given to start with, and then any number of terms can be worked out in succession. If at=I we have a2 =V2.I=v2", a3 =V 2v2, a,= V2V 2V2, etc., or in decimal notation (a..) =1, 1.414, 1·682, 1·834, 1.91 5, . . . Example 6. A similar 'recursive formula', but involving two previous terms, is a..H =0·2a..+t -o·Ia... Here we need to be given at and a. to start with, and then the subsequent terms can be found. For instance, taking at=o, a.=I, we get a3=0·2a.-o·Iat =0·2, a,=0·2aa-o·Ia a=-o·06, a6=0·2a, -0·Ia3=-0·032, etc. 2. SUCCESSIVE APPROXIMATIONS
One very important way in which sequences make their appearance in practice, is where the numerical solution of some problem is attempted by finding successive approximations. These approximations form a sequence whose terms approach the number which is being sought. To take an elementary example, suppose that it is required to find the numerical value of v'2; that is to say, we are looking for a positive number A with the property that A 2=2. By the usual 'square root process' of elementary arithmetic, we learn how to make a sequence of (better and better) approximations to A. The first approximation given by this process, which we might call at, is I. At the next stage we get a 2 =I·4, then 113=1.-1-1, R,=I·4I4, and so on. None of the terms I, 1·4, 1.41, 1.-1-14, ... of this sequence is equal to A. But they approach or 'tend to' A =V2, in the sense 2
GRAPHICAL REPRESENTATION OF A SEQUENCE
that if we go sufficiently far along the sequence, we get numbers which differ from A by as little as we like, It should also be 2'
1'414
I'
24 281 282 4
1'00
96 400 281 II9 00 II296 60 4
noticed that, in practice, this is all we require. For in any given practical application, in which for some reason the value of V2 is required, all that is really necessary is its value correct to a certain number of decimal places, and this we can secure by working out afinite number of steps of the square root process. It is this idea of a sequence which 'tends to' a limiting value, which we shall discuss in the next paragraphs. 3.
GRAPHICAL REPRESENTATION OF A SEQUENCE
It helps in understanding this notion if we can represent sequences graphically. We shall do this in either of two ways; first we can simply mark the values of al , as, as, ' , , as points on a single axis or scale. It is advisable to write above each point the name of the term to which it is equal (and a given point may correspond to many terms), For example, the
sequence 0,
(1-;)=0, t, i, t, ' , ,is represented in Fig. I 02
03 040506 !
o
"
FIG. I
while the diagram for ((-1)")=-1, I, 3
- I , I, . . .
has only two
SEQUENCES
points, each of which serves for infinitely many terms of the sequence.
o
-I
+1
FIG. 2
Although this representation is very compact, it may be preferable to display the sequence more fully. Our second method of depicting a sequence (a,,) is to draw a 'graph', regarding a" as a function of n. However this graph is not a continuous curve, but consists merely of a succession of isolated points, because a" is supposed defined only for n=I, 2, 3, ... Fig. 3 shows the graph of ((-1)"), for which a1 =-1, a2 =1, a3 =-1, a4 =118 , etc. For this sequence, it is not possible to give a real meaning I
2
w
S
4
·1
FIG. 3
to the value of a" for general (non-integral) values of n-for example we cannot define a3/1=(-1)3/2 without using complex numbers. But even in a case, such as a,,=I-~ (Fig. 4), where n
4
THE LIMIT OF A SEQUENCE
it would be possible to 'fill in' the intermediate values, we refrain from doing so, on the grounds that it is only the integral values of n which interest us at the moment.
x
x
x
x
x
x
x
x
X
n
4
2
6
8
10
FIG. 4
4.
THE LIMIT OF A SEQUENCE
Figures 3 and 4 display very clearly the fact that these sequences are 'tending' to 'limits' or limiting values, as n becomes large. It is clear, for example, that as n increases, (_t}n becomes more and more nearly equal to 0, and that
I-~ becomes nearer and nearer to I. We express this by saying n
that (_t}n tends to
0
as n tends to infinity, and that I-~ n
tends to I as n tends to infinity, respectively. The precise definition of this kind of statement is as follows. Definition. A sequence (an) tends to a limit A as n tends to infinity, if, given any positive number h, however small, we can find an integer N" such that all the terms a. of the sequence after the N"th lie between A-h and A+h. Notation. We write 'an--+-A as n--+-oo' (read 'an tends to A as n tends to infinity'), or sometimes lim an=A. Occasionally the 11--+00
phrase 'as n tends to infinity' is omitted, for shortness; then we should write simply an--+-A, or lim an=A. We can think of the terms an as 'approximations' to A. If certain 'limits of tolerance' ±hare allowed, i.e. if we are satisfied when an lies within h of A 5
SEQUENCES
Ath I- - - - _
.Jfo_ -
-
~- -
- -
-
_______ _
A x Xx Xv A-h ~ - - - - - - -x- - - - - - - - - ~ - - - - - .! x )(
n Nh FIG. 5
on either side, then (according to the Definition), all the terms an after a certain one (which we have called the N"th) must lie within these limits. Further, this must continue to be true, whatever value h has-e.g. if we set a narrower tolerance ±k, say (with k1000. Therefore N,,=IOOO would satisfy the conditions in the definition, for the special value h=O·OOI. This is not enough: we must prove that an N" can be found corresponding to any h. This however is quite easy; take N,,=the first integer greater than I/h. Then it is clear that an lies between I-h and I+h, whenever n>N". Example 2. We prove next that xn---+-oas n---+-oo, if -I<Xo (cf. Example 3, p. 7)·
8
RULES FOR CALCULATING LIMITS
For given any K (which we may assume is positive), we have
n' >K if n>
TK. Although VK will not usually be an integer, we
just take N K=the first integer greater than of our definition is satisfied.
Example 6. xn---+-+oo as n---+-oo, if p.6).
n, and the condition
X>I
(cf. Example 2,
For xR>K provided that n log :¥>log K, and this time log :¥>o, because :¥>I, so that we may divide by log :¥ without reversing the direction of the inequality. Therefore :¥n>K if n>log K/log:¥, and thus we may take N K to be the first integer greater than log K/log:¥.
By analogy, we say that a sequence (an) tends to - 00 as n-4oo (written an---+-- 00 as n---+-oo), if, given any number L, we can find an integer N L such that all the terms an after the N Lth are less than L. This is equivalent to saying that -an---+-+ 00. For instance the sequence (-n 2) in Example z tends to - 00. We often write lim an =+ 00, as an abbreviation for 'an---+-+ 00 as n---+-oo', and similarly lim an=- 00. But it should be remembered that '+00' and '-00' are not numbers, and cannot be treated as if they were. Some reasons for this caution will appear in § 7. To complete the classification of sequences, we say that a sequence which does not (i) tend to a limit A (ii) tend to + 00, nor (iii) tend to - 00, is an oscillating sequence. Such sequences are, from our point of view, 'irregular', although they comprise a very wide class. Examples (3) and (4) (an=(-z)n and an=(-I)n) are both oscillating sequences.
6.
RULES FOR CALCULATING LIMITS
If (an) and (b n) are two sequences, and if an---+-A as n---+- 00, while bn---+-B, then the 'sum' sequence (an+bn)=a1+b 1 , a2 +b2 , ... tends to A+B. Naturally this statement requires rigorous proof, using thE definition of § 4. We shall not, for reasons of space, give this proof. S.S.-B
9
SEQUENCES
but it is really just a precise formulation of the following idea: a.. can be made as near A as we like, by making n large enough; similarly bn can be made as near B as we like. When an is very near to A, and also bn is very near to B, then an+b.. must be very near A +B. That means that we can make a. +b,. as near A +B as we like, by making n large enough.
In the same way we have rules for differences and products: Rule 1. If an---+-A and bn---+-B as n---+-oo, then an+bn---+-A+B, an-b,,---+-A-B, and anb,,---+-AB. Furthermore, if c is any constant, then can---+-cA (this is the special case bn=c of anbn---+-AB). Example 1. Find lim (1+ (1)n). Take an=1 (this defines a constant sequence, see Example 4, p. 7) and bn =(l)". We know that a,,---+-I, and bn---+-o (Example 2, p. 6); therefore a,.+b. =1+ (l)n---+-I +0=1.
Example 2. (I+~)(2-:2)---+-(1+0)(2-0)=2. (We use here I
1
the fact that _=n- 1 and a=n- a both ---+-0 (Example 3, p. 7)). n n Rule 2. If a,,---+-A, and b,,---+-B, as n---+-oo, and if each term bn;co, and also B;co, then an/bn---+-A/B. We omit the proof of this rule. The conditions bn;co and B;co are obviously essential, if the expressions a,.lb n and A IB are to have any meaning.
Rule 3. Let (a,,). (b n) be two sequences. (~1 If an---+-+ 00, or if an---+-- 00, then I/an---+-o. (i~) If an---+-+ 00, and b,,---+-afinite limit B, then an+bn---+-+ 00. (iit) If a,,---+-- 00, and b,,---+-afinitelimitB, then an+bn---+-- 00. (iv) If an---+-+ 00, and bn---+-a positive limit B, then a"b"---+- + 00. (v) If a,,---+-+ 00, and bn---+-a negative limit B, then a"b,,---+-- 00. We shall not give formal proofs of these facts, but the reader should try in each case to understand the general principles involved. For example (i) if an---+- + 00, it means that an is very large, for large values of n; hence lla,. is very near o. In (ti), the term b. 10
RULES FOR CALCULATING LIMITS
is near to B, for large n, while an is very large. Even if B is a large negative number, the sum an +bn can be made positive and indeed as large as we like, by making n large enough. Therefore
an +bn---+- + 00. Example 3.
(n+~'=2' 2~, 31, 4i, ... tends nl
to
+ 00.
For
an=n---+-+oo, while bn=~---+-o. n Example 4. 4-n+(lt---+--00. For this is the sum of a term -n, which tends to -00, with a term 4+W n , which tends to the finite limit 4. Example 5. (n +1)/(n2+1)=n(I+~) /
n2(1+ :2)
=(~)(I+~) / (1+ :2)---+-OXI/I=O, as 1£---+-00. (n2+1)/(3n2+n+l) =1(1+:2) / (1+ 31n + 3:2)---+-1.1/1=1, as n---+-oo.
Example 6.
Example 7. an=(I-n 2 )/(1+2n). Take a factor (-n 2 ) out of the numerator, and a factor 2n out of the denominator; we get an
=(-;)(1-:2) / (1+ 2:). The factors in the two last
brackets both tend to 1as n---+- 00. Therefore, for large n, an will be very nearly equal to ( -;) (in the sense that an /( - ; ) will be very nearly
I).
Therefore, since
-!! clearly ---+-- 00, 2
an---+-- 00 as well. Example 8. Rational functions of n. A polynomial in n is a function of the form c"n"+c"_ln"-l+ ••. +cln+cO' where co, cl , ... ,c" are constants. If c,,~o, h is called the degree of the polynomial, and c"n" is called the leading term. A rational function is the quotient of one polynomial by another; e.g. the functions in Examples 5, 6, 7 are all rational functions of n.
:u
SEQUENCES
Let (an) be a sequence whose nth term is a rational function of c"n"+ c"_ln"-l+ ... + Co ( d ) W . n, say an= d /e d k-l d c", /er=o. e can Write n /en + k-I + ... + 0 this
an = dc"n"/e' { 1 +C"-l!.+ kn c"n
•••
+~c"n~}/ " dk-l { 1 + dIe
n1+ ... +dd,.on/eI}.
The terms in the brackets both tend to 1 as n---+- 00. Thus for large n, an behaves like the quotient of its leading terms, in the sense that the ratio of an to this quotient c"n"/dkn/e tends to 1c 1 as n---+-oo. Thus for example, if hk, then an---+-+ 00 or to -
00,
according as C,,/dk is positive or negative. I t is sometimes useful to notice the following rule. Rule 4. If an---+-A and bn---+-B as n---+-oo, and if ano). Example 4.
(I-~)=O. t. i. 1. t .... is bounded above and 1
below. For 0o for all n. Therefore it must tend to a limit, X, say. Now (x"+l)=X2, x 3 , ••• must have the same limit X (Example 5, p. 14). But (x"+l) can also be obtained by multiplying every term of (x") by the constant x. It follows by Rule I (p. 10) that its limit is xX. Hence xX =X, from which (X-I)X =0, and, since we know that X-I~O, it follows that X =0. This is of course the result which we had already found in § 4 (Example 2, p. 6). 10. THE FUNCTIONS
x", n'
AND
nix"
We first collect for reference some of the results obtained in this chapter.
17
SEQUENCES (i) x"---+o as n-+oo, if -I<Xl, then n'/yR-+o as n-+oo, for any value of s. First, if 5!. and, in general, S2'> k+2. This shows that we can make S2" as large as we like, 2
by making k large enough; in other words the subsequence S2' s" SS, ... of (s,,) tends to + 00. By Rule 5 (p. 14), (s,,)
29
INFINITE SERIES
cannot tend to a finite limit, for if it did, any subsequence of
L: ex>
(s,,) would tend to the same limit. Therefore
lin is
n~l
divergent.
5.
SOME RULES FOR CONVERGENT SERIES
We give next some simple rules.
L: L: L: ,,-1 ex>
Rule 6. If
ex>
u",
v" are two convergent series, with sums
,,=1
,,~1
ex>
S, T respectively, then
L: ,,-1
(u,,+v,,) converges and its sum is S+T.
therefore
(u" +v,,) converges. and has sum 5
+ T.
The other
n=1
cases can be established in a similar manner.
L: CIO
Example 1. The series
x"=x+x 2 +x3+ ••. can be ob-
,,=1
L: CIO
tained by multiplymg the series
x"=I+X+X 2 + ••• by
,,=0
the constant x. Therefore its sum is x.(_I_), if -1<X" from Lx", we find, in the CIO
Example 2. By subtracting
CIO
,,=0
,,=0
L (x"-y") CIO
case that x, y arc both between
-I
and
I,
that
is
"=0
convergent and has sum (I/I-X)-(I/I-Y) =(X-y)/(I-X)(I-y).
L L CIO
Rule 7. If
CIO
u,,'
,,=1
v" are two convergent series, with sums
"=1
S, T respectively, and if u"
Example 3.
2 n /n18
diverges, because Un =2n /n 18 does not
n=1
tend to zero as n~oo (it tends to +00, see p. 18). Warning. Test I can never prove that a series is convergent. It is possible that un~o even for a divergent series, for ex-
L ex>
ample we saw (§ 4 (iii), p. 29) that
I/n is divergent,
n=1
although the nth term
7.
I/n~o
as
n~oo.
THE COMPARISON TEST
One of our main tasks is to find 'convergence tests'; that is,
L ex>
tests which enable us to decide whether a given series
Un is
n=1
convergent or not (without necessarily finding its sum). Test
33
I
INFINITE SERIES
is a very easy preliminaiy test, but it is of no use for proving that a series is convergent; it can only tell us that certain series are divergent. In this paragraph, and in §§ 8, 9, we shall be mainly concerned with series whose terms are positive (or at any rate nonnegative, i.e. positive or possibly zero); we reserve the notations p", q" for the terms of such series. The nth partial sum SrI =Pl
L: p,. of positive terms is an 00
+P2+ ... +p"
of a series
,,=1
increasing sequence. For
S,,+1 =S,,+P"+l >s", all
n. Therefore, by the Fundamental Theorem on p. 16, (s,,) either tends to a finite limit or to + 00. This fact makes series of positive terms easier to deal with.
L: p", L: q" be two 00
Test 2. Comparison Test (first form). Let
00
,,=1
.. =1
series of non-negative terms (i.e. p,,>o, and q,.>o, for all n). Then (i) If p" P.. for n> N, i.e. (P .. ) is an increasing sequence. Since the P.. are all positive, it is impossible for them to tend to zero. Therefore ~P .. diverges, by Test I.
L rlnl is convergent. (n!=I.2.3 ••. (n-r).n ,,-0 00
Example 1.
THE INTEGRAL TEST
if n is a positive integer, and O! =1, by definition.) Take p,,=Iln!, then P,,+lIP,,=nl/(n+I)!=Iln+I----+o as n----+oo. Since this limit L=o is less than I, the series converges. (Note: p" is actually the (n+I)th term of this series, since we start with a term Po. This makes no difference, for the limit of (P"+lIP,,) is the same as that of (p,,+t!P"+l))'
I
00
Example 2.
n 2x" (x>o). Here P,,+I/P,,=x(n+I)2In 2
,,=1
=x(I+~) 2--+ x, as n----+oo. Therefore Ln 2x" converges if XI (and not just for integral values x=n), and satisfying the following conditions: (a) f(x) >0, for x >1, and (b) f(x) is decreasing as x increases from I.
I
00
Example 1. The series
lin' (5)0) is in this class. For
,,=1
p"=f(n), where f(x) is the function IIx'. Conditions (a) and (b) are satisfied. 39
INFINITE SERIES
The graph y =/(x) is a curve of the form suggested in Fig. Let F(t) =
J
12.
:/(X)dX; this is the area under the curve between
y
2
n-I
3
n
)C
FIG. 12
X=I and x=t. Let S,,=Pl+P2+ ... +P.. , the nth partial sum
I
00
of the series
P... The area of the shaded part of Fig.
12
is
.. -1
P2+Pa+ ••• +P.. =S,,-Pl (for the width of each shaded rectangle is I, and their heights are /(2)=P2' /(3)=Pa, etc.). This area is less than the area of the curvilinear region F(n), and F(n) is, in turn, less than the sum of the larger rectangles in Fig. 12. Their heights are PI' P2' ... , P.. -l respectively.
Thus S,,-Pl F(n) (p. 40) we see that a,,=s"_l+p,,-F(n»p,,>o. for all n; i.e. (a,,) is bounded below. We shall prove next that (a,,) is a decreasing sequence. For a,,-a"+l=(s,,-s"+l)-(F(n)-F(n+r)) =-Pn+l+(the area under y=f(x) between x=n and x=n+I). In Fig. 13. the shaded rectangle has area P"+l' and it is clear 42
SERIES WITH POSITIVE AND NEGATIVE TERMS
n
n+1 FIG,I3
that this is less than the corresponding area under the curve. Therefore a,,-a"+l>o, i.e. a,,>a"+l' for all n. It follows from the Fundamental Theorem that (a,,) tends to a finite limit. In the casef(x) =r/x, this limit has the value 0'5772 ... and is denoted by i' (Euler's constant). Thus (r+1+1+ .•. +r/n)-log.n~=o'5772 ... , as n----+oo This means that, although we can give no simple formula for the finite sum r+1+1+ ' .. +r/n, it is approximately equal to log.n+o'5772 ... , for large values of n. ro.
SERIES WITH POSITIVE AND NEGATIVE TERMS. LEIBNIZ'S TEST
We have dealt so far mainly with series "Lp", whose terms p" are all positive. For such a series, the sequence (s,,) of partial sums is an increasing sequence, If, however, we take a series 00
such as r-1+1-1+ ••• = I(-r)"+I/n, which has both ,,=1
positive and negative terms, then the sequence (s,,) will not be increasing. The partial sums of the series mentioned start off 5 1 =r, s2=0'5, 5 3 =0'833 ... , s.=o'5 833 ..• , s5=O'7 833 ' , . In fact, this series is convergent, as we can prove with the help of the following test. 43
INFINITE SERIES
Test 6. (Leibniz's Test, or Alternating Series Test.) The series co
L:
(-I)n+la n=a1 -a9 +a3 -a,+ ••• is convergent provided
"-1
(t) (an) is a decreasing sequence, and (it) an~O as n~oo. (A series whose terms are alternately positive and negative is called an alternating series.) Because (an) is decreasing, an -an+1 > 0 for all n. Consider the sequence (s2n-l) =SI' sa, S5' ... of the odd partial sums of co
L:
(-I)n+l an · This is decreasing, since
"=1
S.n+1 -S2n-1 - -a 2n +a Zn- 1< o.
for all n, and bounded below, because
s.n-l = (a 1-a.) + (aa -ac) +
... + (a.n-a -aan-.) +a2n-1> 0
for all n. Therefore (S.n-1) tends to a linIit, 5', say. Similarly, the sequence (s.,.) =s., sc, s,• ... is increasing. since
s."+I-s.n =aZn+1 -aan+I>0, for all n, and also bounded above, for
s.n =a1- (a.-aa) - ... - (a 2n-a -a 2n- 1) -a.n< ai' for all n. Therefore (s.n) tends to a limit, say 5". Now s.n-l -S2n = a.n, and (a. n), being a subsequence of (an). tends to o. Thus 5' =lim s2n-1 = lim (s.n +a.n) =5" +0, i.e. 5' =5". Denote by 5 the common value 5' =5". Then both the even and the odd terms of the sequence (sn) =SI' S2. sa. sc, ... tend to the same limit 5, and so (sn) itself tends to this limit. Therefore the series converges.
52 S4
',
i
0
S5 S, i
S
, ,
S. I )
I
FIG. 14 co
Example 1. The series
L: (-I)n+l/n=I-l+l-!+ ... is n=1
convergent. For this is the case an=I/n, which certainly satisfies the two conditions (i) and (ii) of Leibniz's test. shall 44
me
ABSOLUTE CONVERGENCE
prove in the next chapter (Example log. 2=0'6931 ....)
I,
p. 67) that the sum is
2 (-I)..+I/(2n-I)=I-l+i-t+ ... co
Example 2.
is con-
.. =1
vergent, since (a ..) =(I/(2n-I)) also satisfies both the conditions (i) and (ii). (we shall see in the next chapter (Example 3, p. 68) that its sum is ~.) II. ABSOLUTE CONVERGENCE
It is convenient at this point to recall the following definition: if x is any number, then Ixl (the 'modulus' or 'absolute value' of x) is the numerical value of x, disregarding its sign. For instance, 1-31 =3, 141 =4, 101 =0. This can be expressed more formally by saying Ixl =x, if x> 0, while Ixl = -x, if x< o. Example 1. If R is a positive number, Ixl o, then a.. = l{u.. +u.. ) =u..' while b.. =l{u.. -u..) =0. If u .. 0. Further, Iu ..1 = a.. +b.. and u .. =a.. -b... Since l::lu..1 is convergent, and a.. I to prove that a series for which
48
POWER SERIES
2: 1t!xn is dict)
lim IUn+llunl =+ 00 is divergent. In particular,
.. =1
vergent for all x, except X=O, and therefore R=o for this series. Naturally a power series with zero radius of convergence has little interest for practical purposes.
2: {-r)n+lxnln=x-: ct)
Example 4. The logarithmic series.
3
n=O
x 8 X' +---+ .•. IUn+llunl=l-xnln+rl=lxl{nln+r)~lxl, as
3
4 Therefore the series converges if Ixlr. Thus the radius of convergence is 1. {We shall see (p. 67) that the sum of this series is log.{r+x).) Example 5. The binomial series. By putting y=r in the binomial theorem (p. 25) we obtain the finite series {r+x)1I n~oo.
=
~ (=)xn.
Now the 'binomial coefficient' (:) can be de-
fined, even when a is not a positive integer, by the usual for) (a) =r. H ow· mu Iae ( a) a{a-r) ... (a-n+r) (if n>o, n 1.2 . . • . n 0 ever, in this case the series
I
(:)xn will be infinite, because
n-O
the coefficients
(=) are all non-zero, unless a is a positive integer
or zero. This infinite series which we get by using a value of a which is not a positive integer or zero is called the binomial series. To investigate its convergence, we observe that IUn+llunl
=I(n~r)x / (n)I=I:~:IIXI~lxl, binomial series
i
as
n~oo.
Therefore the
(:)xn=r+ax+ a{;~r)X2+ ..• converges
n=O
absolutely if Ixl r; in other words its radius of convergence is r. {The sum of this series is in fact (r+x)II, for any value of a. See p. 71.) 49
INFINITE SERIES
Example 6. The sine and cosine series. The two series (i)
"" 2
X2
x' x8
{-I)nx2n/{2n)!=I-_+ ___ + ... , and 2! 41 61 n=O
5 ~ {-I)n+Ix2n+I/{2n+I)!=x-_+ x 8 x___ x' + .•. (ii) ~ ,,=0 3! 51 7!
are closely related to the exponential series. For (i) we find IUn+Ilunl =x2/(2n+I)(2n+2)~0 as n~oo, and again, for (ii), IUn+Ilunl =x2/2n(2n+I)~o as n~oo. Therefore each of these series converges absolutely for all values of x, i.e. they have infinite radius of convergence. (It is shown in text-books on the calculus that the sum of the series (i) is cos x, and the sum of (ii) is sin x; in each case x being in radian measure.)
'"
Example 7. It is possible to prove that every power series 2c"xn n=O
has a radius of convergence R (which may be infinite). However it may not be possible to find R by the ratio test. For example, the series
~cnxn = 1+ (;) +x. + (;) 8+x' + (;)' + . ..
would glVe
IUn+1/unl = mn+llxl, if n is even, and Iun+dunl = 2 nl xl, if n is odd. It is clear that lun+dunl does not tend to a limit as n~ 00, and therefore the ratio test cannot be applied. On the other hand, the series converges absolutely if Ixl < I, by the companson test (Test 2) (for Iuni < Ixl n , all n, and l:lxl n is convergent if Ixl I, the nth term utI does not tend to zero and therefore l:c"xn diverges (Test I). Thus this power series has radius of convergence I. Interval of Convergence. If R is the radius of convergence of the
power series
2""
c"x", then the interval -R<xI-Y, and hence U"/I-Uo, and o>T,,(x) >-lxI2"+1/(2n+I) if xM,.. Therefore all the partial sums Sm after s" (we can take n =M,,+ I if we like, for definiteness) lie between s" -k and s.. +k. If we take k to be very small in relation to the accuracy required, this means that the partial sums Sm for m >n are 'practically' (e.g. up to a certain number of decimal places) indistinguishable from s ... Thus s .. would, up to this number of decimal places, represent the full sum. In this sense it would be possible to calculate the sum S to any reqUired degree of accuracy.
6.
DIRICHLET'S CONVERGENCE TEST
Test 8 (Dirichlet's Test). Let (a ..), (b ..) be two sequences such that (t) The sums t.. =b1 +b s+ .•. +b.. are bounded, i.e. there exists a number H>o such that It..1n) write R m, .. =a ..+1bn+1 +a"+2b"+1+ ••. +ambm. Our ainI is to find an integer M" such that IR.n, ..1M k' If we can do this (for an arbitrary positive k), then it will follow from Cauchy'S
2: IX)
convergence principle that
anb .. converges.
.. -1
By (iii), a,,-+-o as n-+- 00. We take M" to be an integer such that an M". Such an integer M" must exist, by the definition of the statement a..-+-o (p. 5); the reason for the curious number k/2H will appear shortly. Remembering that b ..+l =t..+ l - t.., we have
73
FURTHER TECHNIQUES AND RESULTS
Rln,n =an+1(tn+1 -tn) +an+l(tn+l-tn+l ) + . • • + am_l (tm-l -tm-I) +am(tm -tm-l) = -tnan+1 +tn+l(a n+l -an+l) +tn+l(a n +l-an+3) + ... +tm-l(am-l- am) +amtm. Now an+1-an+l' an+l-a n+s, ... am- l -am are all > 0, because (an) is a decreasing sequence (condi-
tion (ii)). Incidentally, conditions (ii) and (iii) together imply that all the an are> o. Therefore IRm,nl < Itnla n+1 + Itn+ll(an+1-anH) +ltnHI(anH-an+a)+ •.• +ltm+ll(am-l-am)+ltmlam, and since each Itnlo). l a
a
2
J
NS+a
L 00
Prove that the sum of
I/{nl +64) lies between 0' 183 ...
n=l
and 0'193 ... (Take N =6.) 3. Show that exp{o'l) lies between 1'1050 and 1'1052, using the estimate for the remainder of the exponential series given on p. 62. (Take N = 3.) 4. (i) Show that e'» = I +x, with an error which is less than 0'0001 if Ixi < 0'01. (ii) Show that ~ I +x = I -l-tx, with an error which is less than 0'0003, if Ixl < 0'05 (see p. 63). 8(1 -1518)1/S -• 5. VerifythatlOx5'=8'-1518,whence{lIO=5 32 768 Use 4 (ii) above to prove that ~IO lies between 1'5846 ... and 1'5856 ... Xl
Xl
6. (i) Show that 10g(1 -x) = - x - - - - - ... if -1<X I, for all 5. For X= I, cony. 5- I. For x=- I, cony. 50.
m.
co
12. I+3X+ 6xl -\ IOX'+ ... =
L
!(n+I)(n+2)xR.
RadiuscollV.~I.
n=O ChaPte~
Ill:
L 00
Xl 2X' X' x' 2X· 6. (iii) x+---+-+--- + ••. = 6 2 3 4 5
n=1
less n is a multiple of 3. while Cn=(Express I+X+X I as (I-X')/(I-X).) I.
77
I cnX", where Cn=- un-
n
nif n is a multiple of 3. 2
Index Absolute convergt'uce, 46 value, ...,
Achilles, 06 Binouual sene3. -49, lA}, 71 theorem, 25 Boundrd sequence, IS Cauchy's CODvcrgf'nc-e pnnriple, 72
Comparison test, 34. 36 Convergence, 27 absolute', 46 mtcrval ')f, 50 principle, 72
See also Tests of
convtrg~nce
Cosine series, 50 Decunal, rN"llrnnc, 28
DIfferentIation of series, 70 Divergf"nr.f'. 27 Equations, solution by iteration, 20 EstlIDation of renlcunder, 58 Euler's COOS1.l0t, 42 Exponr-ntiai !tCncs, 48, 52, 62, 7a
Firute sequt..nce,
1
Sf'neS,24
Gt:'Ometric senes, 25. 28 Gregory's series, 68 Increasmg sequence, 15 Infinite Sf"qucnce, I
senes,25 Integral tcst, 41 Interval of convergence, 50 ItpratlOD, 20 Lrading term,
I I
LeJbniz's senes, 68 test, « Limit, , Logaritlumc senes, 49,
6,
Monotone sequen~, 15 Multiplication of series, 51
Oscillating sequence, 9 Parhal sum, 27 Power series, 47 dIfferentiation. 70 integration, 64 remamder, 61 Product sene., 51
Ratio test, 38, 47
Rational function, I I RecuITmg decimal. 28 Remainder of a series, 32 estunation of, S8
I,
Sequencc~, 1
bounded, conc;;tant,1 fmite, t ge:lt"ral term of, t graphIcal rcprcS('ntation of, 3 increasing, IS inf..rute, I limit of,-; monotone. J.5 of approximations, 2
oacIllatmg,9 recursively defined, 2 SUb-,13 Series absolutely convergent, 46 alternating, 44 binomial. 49, 6