Consider the series .

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

Problem

Consider the series

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

. (a) Calculate the fourth partial sum

S_{4}

. Express your answer as a rational number. (b) Determine the maximum possible error

|S_{4}-S|

, where

S

denotes the exact value of the infinite series. (c) Write the smallest closed interval that must contain

S

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