# Does the series $\sum\limits_{n=1}^\infty\frac{\ (-1)^n \log(n)}{\sqrt{n}}$ converge?

The following series (OEIS A265162) converge or diverge?

$$\sum_{n=1}^\infty\frac{\ (-1)^n \log(n)}{\sqrt{n}}$$

I have proved that this series diverges absolutely.

I tried to use Leibniz criterion:

1. $$a_n >0$$ definitively.
2. The limit of $$a_n=0$$ (as n tends to infinity).
3. $$\log(n)/\sqrt n >\log(n+1)/\sqrt{n+1}$$ definitively

it's ok?

• That is what you need to show, yes. Best to be explicit about what $a_n$ is from the start. You obviously can't just assert (2) and (3), but have to prove them. – Thomas Andrews Jan 23 '13 at 14:28
• 1) and 2) are obvious; 3) it's proved with some passages and with passage to the limit. But wolfram said: "sum does not converge, ratio test inconclusive, root test inconclusive" – user55114 Jan 23 '13 at 14:33
• How can you prove (3) by passing to the limit, when you are trying to prove it for specific values, or do you just mean the function is strictly dereasing as a continuous function? – Thomas Andrews Jan 23 '13 at 14:36

Having this, it is simple to determine the convergence of the original problem.