Is it possible to find the sum of the infinite series $1/p + 2/p^2 + 3/p^3 + \cdots + n/(p^n)+\cdots$, where $p>1$? Is it possible to find the sum of the series:
$$\frac{1}{p} + \frac{2}{p^2} +\frac{3}{p^3} +\dots+\frac{n}{p^n}\dots$$ 
Does this series converge? ($p$ is finite number greater than $1$)

For $p>1$ the series converges.
Lets take the general geometric series;
$$|x|<1\\a_1(1+x+x^2+x^3+...)=\frac{a_1}{1-x}\\a_1(1+2x+3x^2+4x^3+...)=a_1(1+x+x^2+x^3+...)'=(\frac{a_1}{1-x})'=\frac{a_1}{(1-x)^2}$$
So in our case;
$$a_1=\frac{1}{p}\ \ ,\ \ x=\frac{1}{p}\\\frac{1}{p}+\frac{2}{p^2}+...=\frac{1}{p}\cdot\frac{1}{(1-\frac{1}{p})^2}=\frac{1}{p(1-\frac{2}{p}+\frac{1}{p^2})}=\frac{1}{\frac{p^2-2p+1}{p}}=\frac{p}{(p-1)^2}$$
