If the terms in a series approaches 0 as n grows without bound, how can the series possibly diverge? I don't understand that. I can only assume that the inverse is definitely true though - if the limit does not approach 0 then the series can never converge.

I've been solving the problems very awkwardly and usually incorrectly.

For example:

Infinite series:

n! / 6^n

My logic is that it must diverge simply because for very large n, n! becomes greater than 6^n, even though that's not the case at lower n.

For the infinite sum (2 + (3/n))^n, I just logically concluded that 2 does not matter at large n, than 3/n approaches 0, and (3/n)^n just approaches 0 faster, so in my mind the series has to converge. Something tells me that is not the case, though.

I believe my that my thinking is fine for the first, but wrong for the second. Can it still diverge if terms approach 0 as n grows without bound? An example and why, would be great!

Thank you!