Let T= _ n=1 ^ (-1)^ n-1 1 2n-1 be an alternating series involving the reciprocals of…

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

Problem

Let

T=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{2n-1}

be an alternating series involving the reciprocals of odd integers. (a) Compute the fifth partial sum

S_5=\displaystyle \sum_{n=1}^{5}(-1)^{n-1}\dfrac{1}{2n-1}

. Present your answer as a single simplified fraction. (b) Determine the smallest positive integer

N

such that the absolute error

|T-S_N|

is less than

0.005

. Show all work leading to your value of

N

. (Do not attempt to evaluate the infinite series itself.)