Question

Determine whether the series is absolutely convergent, conditionally convergent, or divergent.

Answer 1

\[\sum_{n=1}^{\infty}(-1)^{n}\frac{ \arctan n }{ n^9 }\]

Answer 2

is it decreasing? do the terms go to zero?

Answer 3

if that is not clear,
\[\lim_{n\to\infty}\arctan(n)=\frac{\pi}{2}\]

Answer 4

and therefore it converges for sure, at least conditionally, as the series alternates and the terms go to zero
that is all you have to check for an alternating series