1. Consider the alternating series 1-+- +.., i.e. Eak where ap = :(-1)k-1 for all k e N. 16 (a) Show that ak is absolutely convergent. (b) Consider the series given by odd and even terms, a2k–1 and a2k. Prove that both of these series converge, and that S = O+ E, where S =E Σ Σ ak, Ө— A2k-1, and E = a2k. k=1 k=1 k=1 In part (c) and (d), consider the series bk given by rearranging the terms of ar as follows: b1 = a1, b3k-1 = a2k; b3k = a4k-1; b3k+1 = a4k+1, for all k E N. (c) Write out the partial sum Š, (where S„ = E-, br).
1. Consider the alternating series 1-+- +.., i.e. Eak where ap = :(-1)k-1 for all k e N. 16 (a) Show that ak is absolutely convergent. (b) Consider the series given by odd and even terms, a2k–1 and a2k. Prove that both of these series converge, and that S = O+ E, where S =E Σ Σ ak, Ө— A2k-1, and E = a2k. k=1 k=1 k=1 In part (c) and (d), consider the series bk given by rearranging the terms of ar as follows: b1 = a1, b3k-1 = a2k; b3k = a4k-1; b3k+1 = a4k+1, for all k E N. (c) Write out the partial sum Š, (where S„ = E-, br).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
1
Transcribed Image Text:1. Consider the alternating series 1 – + - 16 +..
i.e. Eak where aj = (-1)k-1 for all k E N.
4
(a) Show that ak is absolutely convergent.
(b) Consider the series given by odd and even terms, Ea2k–1 and a2k. Prove that both of these
series converge, and that S = 0+ E, where
Σ
e =E
and E = >
S =
ak;
a2k-1,
a2k .
k=1
k=1
k=1
In part (c) and (d), consider the series br given by rearranging the terms of ar as follows:
b1 = a1,
b3k–1 = a2k;
b3k
= a4k-1,
b3k+1
= a4k+1;
for all k e N.
(c) Write out the partial sum S, (where S, = -1br).
k=1
(d) Show that for all n E N,
Šan
Š3n-1 = Š3n
= O2n + En,
- a4n-1,
Š3n+1 = S3n + a4n+1•
Here Om = D- a2k–1 and Em
E-1 a2k denote the partial sums for O and E in part (b).
|3|
(e) Prove that (Šn) → S, i.e.,
> bk = S.
k=1
(Hint: Use part (d) to compute Sm for m e {3n – 1, 3n, 3n + 1}, and take a limit.)
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