Calculate sum $\sum_{n=1}^\infty {n\over3^{n-1}}$

Question

Calculate sum $$\sum_{n=1}^\infty {n\over3^{n-1}}$$The result is $9\over 4$ but I don't know how to get that.

Answer 1

We'll differentiate a geometric series. Suppose $|x|<1$; since $\sum_{n\ge 1}x^n=\frac{x}{1-x}$, $\sum_n nx^{n-1}=\frac{1}{(1-x)^2}$. Now take $x=\frac{1}{3}$.