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.

Sum it from $M-1$ to $1$, i.e., sum it all up backwards.
$$\sum_{k=1}^{M-1}\frac{q^{i-1}}{M-i}=\sum_{k=1}^{M-1}\frac{q^{M-k-1}}{k}=q^{M-1}\sum_{k=1}^{M-1}\frac{q^{-k}}{k}$$
Let $q=r^{-1}:$
$$=r^{1-M}\sum_{k=1}^{M-1}\frac{r^k}{k}\tag{$\star$}$$

Recall the geometric series:
$$\frac{1-x^{M-1}}{1-x}=\sum_{k=1}^{M-1}x^{k-1}$$
Integrate wrt $x$ from $0$ to $r:$
$$\int_0^r\frac{1-x^{M-1}}{1-x}\ dx=\int_0^r\sum_{k=1}^{M-1}x^{k-1}\ dx=\sum_{k=1}^{M-1}\frac{r^k}k$$
Thus, you may rewrite your sum as
$$\sum_{k=1}^{M-1}\frac{q^{i-1}}{M-i}=q^{M-1}\int_0^{1/q}\frac{1-x^{M-1}}{1-x}\ dx$$
From there, you may use integration techniques to derive closed form solutions for some $q,M$.
As demonstrated on this graph.
