# Basic Comparison Test

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

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
Staff Emeritus
Homework Helper
Compare it to $1/n^2$...

Compare it to $1/n^2$...

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

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
Staff Emeritus
Homework Helper
Calculate the limit of the terms and show that the limit isn't 0.

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
Staff Emeritus