$\sum_{n=1}^{\infty}(-1)^nn^a\int_{n}^{+\infty}\frac{\ln x}{x(x+1)}dx$, $a\in\mathbb{R}$ converges or diverges 
Does the series $\sum_{n=1}^{\infty}(-1)^nn^a\int_{n}^{+\infty}\frac{\ln x}{x(x+1)}dx$, $a\in\mathbb{R}$ converge or diverge?

I have proved that $\int_{n}^{+\infty}\frac{\ln x}{x(x+1)}dx=\frac{\ln n}{n}+\frac{1}{n}+O(\frac{ln}{n^2})$
Then, $u_n=(-1)^n\frac{\ln n}{n^{1-a}}+\frac{(-1)^n}{n^{1-a}}+O(\frac{\ln n}{n^{2-a}})$.
If $a<1$ , the series converges.
If $a\geq 1$ , I can't resolve the problem. Could you give me some hints? Thank for helping.


If $a\geq 1$ , I can't resolve the problem. Could you give me some hints? 

From what you wrote, if $a\ge1$, then, as $n \to \infty$,
$$
\left|u_n \right|\sim n^{a-1} \ln n,
$$ giving
$$
\left|u_n \right| \to \infty
$$ and the series is divergent.
