Calculate the interval of convergence.

Question

Sigma from n=1 to infinity of (n!*x^n)/n^n.
Calculus - Steve, Friday, July 29, 2016 at 11:52pm
try using the ratio test. You should be able to show that the series converges for |x| < e

I tried and I got
|x|lim n-infinity of |(n/n+1)^n|
which I get |x|<1

Answer 1

Using the ratio test,

An+1/An =

(n+1)!/(n+1)^(n+1)
--------------------- x
       n!/n^n

= (n+1)!/n! n^n/(n+1)^(n+1) x
= (n+1) (n/(n+1))^n * 1/(n+1) x
= (n/(n+1))^n x
Hmmm. what to do?
= ((n+1)/n)^-n x
= (1 + 1/n)^-n x
This should look familiar
(1 + 1/n)^-n --> 1/e
So, now we have the ratio test saying that

An+1/An = 1/e x

For the series to converge, the ratio must be less than 1, which means that

|x| < e