Why is the infinite series n^n/(n+1)! and n^(n+1)/n! divergent by the ratio test?

Question

Why is the infinite series nⁿ/(n+1)! and n⁽ⁿ⁺¹⁾/n! divergent by the ratio test? I see how it is divergent by the test for divergence, but I am supposed to prove it through the ratio test. Every time I do the ratio test, I get 1, which would mean the ratio test would be inconclusive. Can someone please help or show me how it is suppose to be proven?

Answer 1

The ratio test involves a_n+1/a_n.

In the first case the ratio is [(n+1)ⁿ⁺¹ * (n+1)!]/[(n+2)! * nⁿ] = [(n+1)/(n+2)] * [1+(1/n)]^n.

^{The limit of the ratio is e>1 and the series diverges.}

^{You try the second one yourself in a similar way.}