Summation Formula for Series I have a series of the form :
\begin{equation}
\frac{1}{M-1} + \frac{q}{M-2} + \frac{q^2}{M-3} + \frac{q^3}{M-4} + \frac{q^4}{M-5}+\dots = \sum_{i=1} ^{M-1} \frac{q^{i-1}}{M-i}
\end{equation}
I want to solve this series to find a general formula that provides its sum. I am not able to figure out the best and easy way to proceed with this. I would be glad if anybody could point me the right direction for solving such series.

Your series may be written as
$$
\sum_{n=0}^\infty \frac{q^n}{M-n-1},\quad |q|<1, \,M \neq 1,2,\cdots,
$$ this is an instance of the Lerch transcendent  function $\Phi$: $\Phi(q,1,1-M)$. For general parameters $q$ and $M$, there is no known simple closed form of it.
