# HELP! Stuck on an Infinite Series. Thanks

I have the infinite series...

$\sum n^n/n!$

somehow, I need to use the Ratio Test... and shoe that the resulting limit
is equal to e. I can't figure out what I am doing wrong algebraically.

Right now I have simplified this limit to lim((n+1)^n/n^n).

Thanks.

Last edited by a moderator:

THE SERIES IS n^n/n!

Mark44
Mentor
I have the infinite series...

$\sum n^n/n!$

somehow, I need to use the Ratio Test... and shoe that the resulting limit
is equal to e. I can't figure out what I am doing wrong algebraically.

Right now I have simplified this limit to lim((n+1)^n/n^n).
This is the same as (1 + 1/n)n.

To evaluate a limit like this, let y = (1 + 1/n)n. Now take the ln of both sides, and then take the limit as n -> infinity. Your textbook should have an example of this type of limit.

I know how to evaluate this type of limit, my problem is algebraically getting to the point of writing lim((n+1)^n/n^n)=lim(1+1/n)^n. How do I go from my ratio test to seeing the limit as equal to the latter limit?

Thanks

Mark44
Mentor
It's fairly basic algebra.
$$\frac{(n + 1)^n}{n^n} = \left(\frac{n+1}{n}\right)^n$$

Wow, third shift is really getting to me. Thanks Alot.

You should all ready know this property of exponents:

$\frac{a^{n}}{b^{n}}$=($\frac{a}{b}$)$^{n}$

Bleh...too late. I fail at tex. lol