Mathematics

Problem

Investigate the convergence of the series

T=\sum_{n=1}^{\infty}\frac{\sin\!\left(\dfrac{1}{n}\right)}{n^{3/2}}.

Respond to each part below. (a) Prove that

0<\sin x < x

for all real numbers

x

satisfying

0<x<\dfrac{\pi}{2}

. Present a complete argument (algebraic, geometric, or calculus-based). (b) Use the inequality from part (a) with

x=\dfrac{1}{n}

to obtain an explicit upper bound for

\dfrac{\sin(1/n)}{n^{3/2}}

in terms of a power of

n

. (c) Compare the bound found in part (b) to a well-known

p

-series. Determine whether

T

converges or diverges and justify your conclusion rigorously. (d) State whether the series

T

converges absolutely. Support your answer with appropriate reasoning.