I'm doing some review of series and using the ratio test for convergence.
I have the problem (the summation of) n!x^n/n^n
I remember the ratio of convergence = e but I forgot the trick to get to there. Would someone mind refreshing me?
Thanks!
I'm doing some review of series and using the ratio test for convergence.
I have the problem (the summation of) n!x^n/n^n
I remember the ratio of convergence = e but I forgot the trick to get to there. Would someone mind refreshing me?
Thanks!
Ratio test -> $\displaystyle \lim_{n \to \infty} \bigg{|}\frac{a_{n+1}}{a_n}\bigg{|}$
So we get...
$\displaystyle \lim_{n \to \infty}\bigg{|}\frac{(n+1)!x^{n+1}n^n}{n! x^n (n+1)^{n+1}}\bigg{|}$
$\displaystyle = \lim_{n \to \infty}\bigg{|}\frac{n!x^{n+1}n^n}{n! x^n (n+1)^n}\bigg{|}$
$\displaystyle = \lim_{n \to \infty}\bigg{|}x\frac{n^n}{(n+1)^n}\bigg{|}$
$\displaystyle = \lim_{n \to \infty}\bigg{|}x\bigg{(}\frac{n}{n+1}\bigg{)}^n\bi gg{|}$
$\displaystyle = \lim_{n \to \infty}\bigg{|}x\bigg{(}1 + \frac{1}{n}\bigg{)}^n\bigg{|}$
$\displaystyle = |x e^{-1}|$