Does the series with terms n/e^n converge or diverge

[Ki-nana18]

Homework Statement

Does this series converge or diverge? infinity$$\Sigma$$n=1 (n!/e^n)

Homework Equations

The ratio test.

The Attempt at a Solution

lim n--> infinity ((e^n)(n+1)!)/(e^(n+1))

I don't know what to do from here...

 

[Mark44]

You're missing the n!. Set up the ratio of a_(n+1)/a_n, simplify it, then take the limit.

 

[Ki-nana18]

This is what I have:

lim n-->infinity [(n+1)!/(e^(n+1))][(e^n)(n!)]
lim n-->infinity [(n+1)/(e)]

Did I simplify right? And do I just plug in infinity now?

 

[Mark44]

Yes, that's simplified correctly. No, you don't actually plug in infinity, but as n gets larger and larger, what happens to (n + 1)/e?

 

[Ki-nana18]

It also gets larger. Which means that the series diverges.

 

[Mark44]

Yes and yes. Using the ratio test, you found that lim a_(n + 1)/a_n is infinity, and so the ratio test tells us this series diverges.

 

[Ki-nana18]

Yay! Thank you!