How to determine whether the following two infinite series converge absolutely, converge conditionally, or diverge.

Question

I need some guidance on how to solve these, I'm not understanding series and sequences too well and I need an explanation that hasn't come from my lecturer.

$$\sum_{k=1}^\infty \frac{\log k}{k^2}$$

$$\sum_{k=1}^\infty (-1)^{k-1}\frac{2^k}{2^k+k^2}$$

Answer 1

HINT

For the first one let consider

$$\sum \frac{1}{k^\frac32}$$

that diverges or converges? which kind of test can we use?

For the second $\sum (-1)^{k-1}a_k$ what about $$\lim_{k\to \infty} a_k$$