Limit of $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$ What is the limit of this sequence $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$?
Where $a$ is a constant and $n \to \infty$.
If answered  with proofs, it will be best. 

Let $f(x) = 1+x + x^2+x^3+ \cdots$. Then the radius of convergence of $f$ is $1$, and inside this disc we have $f(x) = \frac{1}{1-x}$, and $f'(x) = 1+2x+3x^2+\cdots = \frac{1}{(1-x)^2}$.
Suppose $|x|<1$, then we have $xf'(x) = x+2x^2+3x^3+\cdots = \frac{x}{(1-x)^2}$. 
If we choose $|a| >1$, then letting $x = \frac{1}{a}$ we have $\frac{1}{a} + \frac{2}{a^2}+ \frac{3}{a^3}+ \cdots = \frac{\frac{1}{a}}{(1-\frac{1}{a})^2}= \frac{a}{(a-1)^2}$.
