TITLE: Evaluate the series $\sum _{n=1}^{\infty} \frac{n}{5^n}$
QUESTION [5 upvotes]: $$\sum _{n=1}^{\infty}\frac{n}{5^n}$$
I tried to plug in $n=1,2,3,4,...$ but I can't use common ratio to solve problem.
I think there is another way like using differentiation or integral but I don't no exactly what to do.

REPLY [5 votes]: $$S=\sum_{n\geq 1}\frac{n}{5^n}$$
is clearly absolutely convergent. We have
$$\begin{eqnarray*} 4S = 5S-S &=& \sum_{n\geq 1}\frac{n}{5^{n-1}}-\sum_{n\geq 1}\frac{n}{5^n}\\&=&1+\sum_{n\geq 1}\frac{n+1}{5^n}-\sum_{n\geq 1}\frac{n}{5^n}\\&=&1+\sum_{n\geq 1}\frac{1}{5^n}\\&=&1+\frac{\frac{1}{5}}{1-\frac{1}{5}}=1+\frac{1}{4}=\frac{5}{4} \end{eqnarray*}$$
hence it follows that $\color{red}{\large S=\frac{5}{16}}$.