Convergence of \sum_{n=1}^\infty \frac{1}{n!}: A Basic Comparison Test

[PCSL]

\sum_{n=1}^\infty \frac{1}{n!}

I understand what n! means, but I have no clue what to compare this to. It is obvious to me that the sum converges, but I'm not sure how to prove it. I assume I would compare it to a p-series but I need help. Thanks!

 

[micromass]

Compare it to 1/n^2...

 

[PCSL]

Compare it to 1/n^2...

lol, I just realized how simple this is. my bad.

 

[PCSL]

Sorry, one more.

\sum_{n=1}^\infty \frac{2^n}{n^2}

What would I compare this to?

I can clearly see that it diverges since numerator is waaaay bigger but I don't know how to prove it.

 

[micromass]

Calculate the limit of the terms and show that the limit isn't 0.

 

[PCSL]

Calculate the limit of the terms and show that the limit isn't 0.

Sorry I didn't specify. I understand how to use the limit test. For this problem I am supposed to compare it to something. Thanks for putting up with my questions :).

edit: since 2^n is soo much bigger then n^2 can I compare it to 2^n?

 

[micromass]

Maybe the harmonic series??

It's a stupid exercise anyway if you're not allowed to do the limit test.