Mathematics

Problem

Consider the series

W=\sum_{n=1}^{\infty}\frac{\arctan\!\left(\dfrac{1}{n}\right)}{n}.

Respond to the following prompts. (a) Explain why

0<\arctan\!\left(\dfrac{1}{n}\right)\le \dfrac{1}{n}

for every positive integer

n

. Provide a brief justification. (b) Use the inequality from part (a) to obtain an upper bound for the

n

th term of

W

that involves a

p

-series expression. (c) Determine whether the series

W

converges or diverges. Support your answer with clear reasoning that references the bound established in part (b).