Direct Comparison test of 1/n! to what?

What should I compare if the question requires me to prove its convergence/divergence using the direct comparison test. I know it converges. It seems like for , and converges by p-series test. The first four terms are obviously finite. Is that enough? Is there a better series to compare it to?

Hey MSUMathStdnt.

The p-series comparison should be more than sufficient.

How do you prove that 1/n! < 1/n^2?

Prove n! > n^2 for all appropriate n (i.e. n > 3).

which is the same as proving (n-1)! > n. induction is your friend, here.

or you can just use "brute force":

(n-1)! > (n-1)(n-2) = n² - 3n + 2 > n² - 3n = n(n - 3) ≥ n, if n ≥ 4 (so that n - 3 ≥ 1).

Or, what amounts to the same thing as Deveno did, if ,

.

- Hollywood

How come some threads have a little "thanks" link in the posts of people who reply to you and some (like this one) don't?
P.S. Thanks for replies, guys (and gals, if any).

Some guesses:

1. You need to be logged in - in fact, I think you need to be logged in when you download the page.

2. You can not give thanks to yourself.

3. You have to mouse over the post.

It's working for me on this thread now - I've never noticed that it's not working.

- Hollywood