Mathematics

Problem

Let

T(k)=\sum_{n=2}^{\infty}\frac{(\ln n)^{k}}{n^{2}}, \qquad k\in\mathbb R.

Investigate the convergence properties of

T(k)

. (a) Rewrite the general term

t_n=\dfrac{(\ln n)^{k}}{n^{2}}

in a form that isolates a power of

n

multiplied by a slowly varying factor. (b) Determine the set of real numbers

k

for which the series

T(k)

converges. (c) For any

k

outside the set found in part (b), prove that

T(k)

diverges.