Need help proving convergence of this series

[swampwiz]

n! / nⁿ

I have proven that the sequence converges numerically, but I can't do it analytically, and can't do anything for the series (maybe it the series doesn't converge?)

 

[aesir]

Can you find a relation between the n-th term and the n+1-th term?

Or if you want the asymptotic behavior you can use the stirling's approximation

 

[dimension10]

Use the ratio test

 

[Skrew]

First note its a decreasing sequence bounded by 0, so by the monotone convergence thereom it converges to some number.

If the series is going to converge, the sequence needs to go to 0, which it does as n!/n^n <= 1/n^2 (so use the squeze theorem).

By the p test(you could also use ratio test), we know the series where x_n = 1/n^2 converges.

Thus by the comparision test we see that the series with x_n = n!/n^n converges.