How I can calculate $\sum_{n=0}^\infty{\frac{n}{3^n}}$? For more than one series of any search function known not seem to find the sum of this series:
$$\sum_{n=0}^\infty{\frac{n}{3^n}}$$
I've found that converges with the quotient criterion.
Could you give me some suggestions for finding the sum of this series?
Thanks!

Since $0<a= \frac{1}{3}<1$ the function $a^n$ converges uniformly, so you can interchange differentiation/integration and summation. Rewrite the expression as 
$$
a\sum_{k=0}^{\infty}\frac{da^k}{da}=a \cdot \frac{d}{da}\sum_{k=0}^{\infty}a^k
$$
Can you  handle from here? 
