Considering $ x \in \mathbb{R} \setminus\{0\}$, I want to study: $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{x^n}n\left(\frac{\pi}{2}-\arctan(n)\right)\log\left(2+\frac{1}{n}\right)$$
I want to find for which the set of $x$ where the series converge and whether to not it converges uniformly on $(1,\infty)$.
I was trying to use Mertens theorem for product of series, without success because the series with only $\frac{\pi}{2}-\arctan(n)$ diverges. What should I try?
Note that $$\lim_{x\to\infty}\frac{\frac{\pi}{2}-\arctan x}{\frac{1}{x}}=\lim_{x\to\infty}\frac{-\frac{1}{1+x^2}}{-\frac{1}{x^2}}=1,$$ therefore $n\left(\frac{\pi}{2}-\arctan n\right)\to 1$ as $n\to\infty$. Moreover, $\log\left(2+\frac{1}{n}\right)\to\log 2$, which means that for the general term of the series to converge to $0$, we should have that $(-1)^nx^{-n}\to 0$, therefore $|x|>1$. On the other hand, the series converges absolutely when $|x|>1$, so the series converges if and only if $|x|>1$.
For uniform convergence, note that the series cannot be uniformly Cauchy (check the behavior close to $1$), so the series is not uniformly convergent in $(1,\infty)$.