I need to find out if:

Summation of n = 1 to infinity of n^n/n!

converges or diverges. If it converges, what is the sum?

I tried the ratio test, since it has a factorial in it, and I end up getting ((n+1)/n)^n, where I apply L'Hopital's Rule, by taking the natural log, pulling the n forward and whatnot, but I'm not sure where to go from there. I mean, I'm sure that I can make it a fraction, but taking limit as n goes to infinity of ln((n+1)/n)/(1/n), and since it's in the 0/0 form, I can take the derivatives, and then take e to that answer, and if it's less than one, it converges. Greater than one is divergence, and equal to one is inconclusive.

However, would the ratio found be the sum that the problem is looking for (if it converges)?

Am I approaching this correctly?