For what values of $p$ is this series absolutely / conditionally convergent?

Question

For what values of $p$ is this series

$$\sum_{k=1}^{\infty} {(-1)^k}{k^p}{logk}$$

                 a) absolutely convergent 

                 b) and conditionally convergent?

I think that if $p ≤-2$ then it will be absolutely convergent and if $p≤1$ then it will be conditionally convergent.

Please tell me if it's correct or not or give me any hints.

Answer 1

Let let $p=-n$ then

$$\sum_{k=1}^{\infty} {(-1)^k}\frac{logk}{k^n}$$

Therefore the the series

• converges absolutely for $n>1$, that is $p<-1$,
• and it is conditionally convergent for for $n>0$, that is $p<0$.