ratio test on $\sum_{n=1}^{\infty} \frac{n}{5^n}$ $\sum_{n=1}^{\infty} \frac{n}{5^n}$

doing the ratio test i get

$ \frac{n}{5^n} * \frac{5^{n+1}}{n+1} $

which results in 

$ L = 5 \frac{n}{n+1} $

as limit reaches infinity on $L$. i get that $L$ is greater than 1 which is undefined behavior. but wolfram confirms it is 5/16. help
You divided thew wrong way around:
$$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\lim_{n\to\infty}\frac{n+1}{5^{n+1}}\frac{5^n}{n}=\lim_{n\to\infty}\frac15\frac{n+1}{n}=\frac15&lt;1$$
So, by the ratio test, the series converges.
Using Cauchy root test, gives us the following:

$$\lim_{n\to\infty}\sqrt[n]{\frac{n}{5^n}}=\frac{1}{5}\lim_{n\to\infty}\sqrt[n]{n}=\frac{1}{5}&lt;1$$

Thus, the given series is converges.  

Edit: Yes, I know that the OP asked the ratio test.