How can I solve this infinite sum? I calculated (with the help of Maple) that the following infinite sum is equal to the fraction on the right side.
$$
\sum_{i=1}^\infty \frac{i}{\vartheta^{i}}=\frac{\vartheta}{(\vartheta-1)^2}
$$
However I don't understand how to derive it correctly. I've tried numerous approaches but none of them have worked out so far. Could someone please give me a hint on how to evaluate the infinite sum above and understand the derivation?
Thanks. :)

let $$S=\sum_{i=1}^\infty\frac{i}{\theta^i}, (\theta>1),$$
then
$$
\begin{align}
S-\frac{1}{\theta}S&=\sum_{i=1}^\infty\frac{i}{\theta^i}-\sum_{i=1}^\infty\frac{i}{\theta^{i+1}}\\
&=\sum_{i=1}^\infty\frac{i}{\theta^i}-\sum_{i=2}^\infty\frac{i-1}{\theta^{i}}\\
&=\frac{1}{\theta}+\sum_{i=2}^{\infty}\frac{1}{\theta^i}\\
&=\frac{1}{\theta}+\frac{1}{\theta^2-\theta},
\end{align}
$$
which yields
$$S=\frac{\theta}{(\theta-1)^2}$$
