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 $1$)

Let:
$
S = {1\over p} + {2\over p^2} + {3 \over p^3} + \cdots \\
\implies {S\over p} = {1\over p^2} + {2\over p^3} + {3\over p^4} + \cdots.
$
Hence,
$
S\left ( 1 - {1\over p} \right ) = {1\over p} + {1\over p^2} + {1\over p^3} + \cdots \\
S\left ( 1 - {1\over p} \right ) = \frac{1}{p}\frac{1}{1-{1\over p}} = \frac{1}{p-1}\\
\implies S = \frac{p}{(p-1)^2}.
$
The series converges for $p>1$.
