Find the radius of convergence of Σ (n^n / n!)(x)^n for (n=1)

Please show all working, thank you!

the ratio test will work here. it states that a series converges if |a_(n+1)/a_n| approaches a value less than 1 and will diverge if is greater than 1. if it equals 1, we do not know whether it converges or diverges. but since we are finding the radius of convergence, we do not care whether it converges/diverges for the finite number of values x can take on such that the value is equal to 1.

|a_(n+1)/a_n|
= |[(n + 1)^(n+1) / (n + 1)! * x^(n+1)] / [n^n / n! * x^n]|
= |[(n + 1)^(n+1) * n! * x^(n+1)] / [(n + 1)! * x^n * n^n]|
= |[(n + 1)^(n+1) * x] / [(n + 1) * n^n]|
= |(n + 1)^n * x / n^n|
= |((n + 1)/n)^n * x|
= |(1 + 1/n)^n * x|

and that approaches |e * x| = e*|x| as n approaches infinity. in order for that to be less than 1, we must have:

e*|x| < 1
|x| < 1/e

so the radius of convergence is 1/e.