Suppose we have a functional series $$\sum_{n=1}^\infty \frac{(-1)^n n^x}{|x|^n}$$ and need to determine whether its sum is bounded on $x \in (-\infty, -1)$
I think the task doesn't imply to calculate this sum. As I understand if I can show that this series is less than other, easy-to-calculate series (infinite geometric series, for example) then I can conclude that the sum is bounded. But what can I do in order to prove that the sum isn't bounded?
Do you know any tricks for such problems? Thanks in advance for any hints!