Hello

Here, I'd split the series into and show that each one converges absolutely/alternately.

To apply the alternating test you have to show, for example for the first series, that decreases (with ) :

for sufficiently large.

Given that

The RHS tends to 0 as approaches , the squeeze theorem tells us does the same and you can conclude.

Note : this series converges absolutely so I wrote this because you asked it but showing directly absolute convergence is quicker.

for sufficiently large and converges absolutely by comparison to the exponential series... (the same idea applies for the other part of the series, compare it to an exponential series)Problem (2): lets assume we figured out problem (1) which test would I use afterward to see if it converges absolutely?

Shoudn't the series start with and not 1 ?Does this series converge absolutely or conditionally:

Problem (2): Which test do I use to see if it converges absolutely? I tried using comparison test:

converges (p-series with p>1) but this has no conclusion on whether converges or not.

decreases for so

We get

Substituting , one can show that the integral tends to when n approaches ... hence, the series doesn't converge absolutely.