# Thread: ratio test for convergence

1. ## ratio test for convergence

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!

2. Originally Posted by morganfor
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!
for a series $\displaystyle \sum_{n=0}^{\infty} {a_n}$ you have,

$\displaystyle X = lim_(n \rightarrow \infty) |\frac{a_{n+1}}{a_n}|$

Ratio test will tell you that:

if X<1, the series is absolute convergent
if X>1, the series diverges

3. Originally Posted by morganfor
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...

$\displaystyle \lim_{n \to \infty}\bigg{|}\frac{(n+1)!x^{n+1}n^n}{n! x^n (n+1)^{n+1}}\bigg{|}$

$\displaystyle = \lim_{n \to \infty}\bigg{|}\frac{n!x^{n+1}n^n}{n! x^n (n+1)^n}\bigg{|}$

$\displaystyle = \lim_{n \to \infty}\bigg{|}x\frac{n^n}{(n+1)^n}\bigg{|}$

$\displaystyle = \lim_{n \to \infty}\bigg{|}x\bigg{(}\frac{n}{n+1}\bigg{)}^n\bi gg{|}$

$\displaystyle = \lim_{n \to \infty}\bigg{|}x\bigg{(}1 + \frac{1}{n}\bigg{)}^n\bigg{|}$

$\displaystyle = |x e^{-1}|$