Consider the infinite series S= _ n=1 ^ n n^ 2

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

Problem

Consider the infinite series

S=\sum_{n=1}^{\infty}\frac{\sin n}{n^{2}}.

Answer the following questions. (a) Verify the inequality

|\sin n|\le 1

for every positive integer

n

. State why this inequality holds. (b) Use the result in part (a) to obtain an upper bound for

\left|\dfrac{\sin n}{n^{2}}\right|

that involves a familiar

p

-series term. (c) Decide whether the series

S

converges absolutely, converges conditionally, or diverges. Provide a clear justification based on your bound from part (b).