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!

Let $T=\sum_{n=0}^\infty\frac1{3^n}$, so $3T=3+\sum_{n=1}^\infty\frac1{3^{n-1}}=3+T$, and $T=3/2$.
Now let $S=\sum_{n=0}^\infty\frac n{3^n}$. So $$S/3=\sum_{n=0}^\infty\frac{n+1-1}{3^{n+1}}=\sum_{n=0}^\infty\frac{n+1}{3^{n+1}}-\sum_{n=0}^\infty\frac{1}{3^{n+1}}=(S-0)-(T-1),$$ and 
 $$ S=\frac34. $$
(If needed, the manipulations with infinite series can be easily formalized by replacing them with manipulations with partial sums, and taking limits.)
