$$ \sum_{n = 1}^\infty \frac{(-1)^n}{n^x \ln(n + x)} $$
Am I right that this series converging absolutely, while $ x \in(0;\infty) $ and don't converge on $(-1;0]$? (I don't consider complex value of logarithm).
I have hard times proving that so any hints or ideas are welcomed.
By alternating series test the given series converges for $x\ge0$ since $\frac{1}{n^x \ln(n + x)}\to 0$ and is decreasing.
The series converges absolutely for $x>1$ by limit comparison test with $\sum \frac 1{n^x}$.
For $x<0$ the series diverges since $|a_n|\to \infty$.