Consider the series S= _ n=1 ^ (-1)^ n+1 n^3

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series

S=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3}.

A new series

T

is obtained by writing the positive terms of

S

in their original order and, immediately after each positive term, inserting the next negative term of

S

. (a) Show that

S

is absolutely convergent. (b) Explain why

T

converges. (c) Describe, without calculating any numerical value, the relationship between

\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3}

and

\displaystyle\sum_{k=1}^{\infty}b_k

, where

\{b_k\}

denotes the terms of

T

. Give a clear justification based on your work in parts (a) and (b).