# Math Help - series convergency

1. ## series convergency

Hello, I am looking at the following series:

sum, n=1 to infinity
{n^3} / e^n

using the ratio test, we get to:
{(n+1)^3} / en^3
however I cannot understand the final step which goes to 1/e which less than 1, so proves convergency.

Can someone please show me how it goes from :
{(n+1)^3} / en^3
to
1/e

Thanks kindly.

2. ## Re: series convergency

Originally Posted by fran1942
sum, n=1 to infinity
{n^3} / e^n
$\lim_{n\to\infty}\left|\frac{(n+1)^3/e^{n+1}}{n^3/e^n}\right|$

$=\lim_{n\to\infty}\frac{(n+1)^3e^n}{n^3e^{n+1}}$

$=\frac1e\lim_{n\to\infty}\frac{(n+1)^3}{n^3}$

$=\frac1e\lim_{n\to\infty}\left(\frac{n+1}n\right)^ 3$

$=\frac1e\lim_{n\to\infty}\left(1+\frac1n\right)^3$

$=\frac1e < 1$