- #1

Masaki

- 8

- 0

## Homework Statement

Determine (absolute/conditional) convergence or divergence of the series

[itex]\displaystyle\sum_{n=1}^{\infty} (-1)^n \left(\frac{\pi}{2}-\text{arctan}({\log{n}})\right)[/itex]

## The Attempt at a Solution

It is easy to see that the series is (conditionally) convergent by Leibniz's rule, but I am not sure how to deny the absolute convergence of the series. To use the comparison test, I must show, for example,

[itex]f(x) = \displaystyle\left(\frac{\pi}{2}-\text{arctan}(\log{x})\right) - \frac{1}{x} > 0[/itex]

for all [itex]x \geq 1[/itex], which seems quite difficult (in fact, I have no idea how to prove it). Is there any other, better way to show that the series is not absolutely convergent?