I have a few problems concerning infinite series which I need some help on. I'm supposed to determine the convergence/divergence using ONLY the comparison tests. Here are the problems:

For the first and the third ones, I want to apply the plain comparison test, using as the series which I compare it to. However, I need a way to prove that is greater than for all n [1, . Once I prove this, how would I apply it to the third problem since it's ? By the way, that is e^n^2...it's kind of hard to tell with the latex. For the second problem, here is what I have so far:

So, now using the limit comparison test...

= =1

The limit exits, is finite, and is greater than zero...so the original series will behave the same as but I'm not sure how to prove whether this series converges/diverges? Any help would be appreciated.