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 $1$)

The exact value is $\frac{\frac{1}{p}}{(1-\frac{1}{p})^{2}}$ when $-1<\frac{1}{p}<1$. Since $p>1$ in this case, the series converges.
Proof:
$$
\frac{1}{p}+\frac{2}{p^{2}}+\frac{3}{p^{3}}...
$$
$$
=(\frac{1}{p}+\frac{1}{p^{2}}+\frac{1}{p^{3}}...)+(\frac{1}{p^{2}}+\frac{1}{p^{3}}+\frac{1}{p^{4}}...)+(\frac{1}{p^{3}}+\frac{1}{p^{4}}+\frac{1}{p^{5}}...)...
$$
$$
=\frac{\frac{1}{p}}{1-\frac{1}{p}}+\frac{\frac{1}{p^{2}}}{1-\frac{1}{p}}+\frac{\frac{1}{p^{3}}}{1-\frac{1}{p}}...
$$
$$
=\frac{\frac{\frac{1}{p}}{1-\frac{1}{p}}}{1-\frac{1}{p}}
$$
$$
=\frac{\frac{1}{p}}{(1-\frac{1}{p})^{2}}
$$
