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. 

The limit does not exist if $|a|\le 1$, and does if $|a|\gt 1$. The fact that it exists if $|a|\gt 1$ can be shown using the Ratio Test.
So we concentrate on the value of the limit, when it exists.
Let $S_n$ be our sum, and let $S$ be its limit.  Then 
$$aS_{n+1}-S_n=1+\frac{1}{a}+\frac{1}{a^2}+\cdots+\frac{1}{a^n}.$$
Let $n\to\infty$. The left-hand side approaches $(a-1)S$, while the right-hand side approaches the sum of an infinite geometric series. That sum is $\frac{a}{a-1}$. Thus $S=\frac{a}{(a-1)^2}$. 
