Infinite series

[corruptedmage25]

how would u test for convergence for infinite series of

n!/n^n

 

[jondbond007]

You prove convergence with the ratio test. You find the limit as n->infinity of the series(n+1) over the series(n).

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

(n+1)!/n! * n^n/(n+1)^(n+1)

(n+1)*(n+1)^-1 * (n/(n+1))^n

and the limit as n approaches infinity of (n/(n+1))^n is 1/e - its the inverse of the definition of e

(1/e)<1 therefore convergent

hope my work is easy enough to follow