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. 

With $S_n = \frac{1}{a} + \frac{2}{a^2} + \frac{3}{a^3} + \cdots \frac{n}{a^n}$ and using the closed form of geometric sums,
$$
\begin{align}
\lim_{n \to \infty} S_n &= \frac{1}{a} + \frac{2}{a^2} + \frac{3}{a^3} + \cdots \\
&= \frac{1}{a} + \frac{1}{a^2} + \frac{1}{a^3} + \cdots \\
& \phantom{=\frac{1}{a}} + \frac{1}{a^2} + \frac{1}{a^3} + \cdots \\
& \phantom{=\frac{1}{a} + \frac{1}{a^2}} + \frac{1}{a^3} + \cdots \\
& \phantom{=\frac{1}{a} + \frac{1}{a^2} + \frac{1}{a^3}} \ddots \\
&= \frac{1}{a - 1} + \frac{1}{a} \cdot \frac{1}{a - 1} + \frac{1}{a^2} \cdot \frac{1}{a - 1} + \cdots \\
&= \frac{1}{a - 1} \left( 1 + \frac{1}{a} + \frac{1}{a^2} + \cdots \right) \\
&= \frac{1}{a - 1} \cdot \frac{a}{a - 1} \\
&= \frac{a}{(a - 1)^2}
\end{align}
$$
