AP

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

Problem

Let

S=\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{n^{3}+1}.

(a) Establish an inequality of the form

0<\dfrac{n}{n^{3}+1}\le c\,\dfrac{1}{n^{2}}

that is valid for all

n\ge N

, where

c

and

N

are positive constants you must determine. (b) Use part (a) to decide whether

S

converges or diverges, providing complete justification. (c) Compute the least integer

k

such that the partial sum

\displaystyle\sum_{n=1}^{k}\dfrac{n}{n^{3}+1}

approximates

S

with an error whose magnitude is less than

0.001

. Explain your reasoning.