HELP Stuck on an Infinite Series. Thanks

[danerape]

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).

Please Help...


Thanks.

 

[danerape]

THE SERIES IS n^n/n!

 

[Mark44]

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.

 

[danerape]

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]

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

 

[danerape]

Wow, third shift is really getting to me. Thanks a lot.

 

[Ivan92]

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