Show $\sum_{j=0}^\infty (j+1)m^{-js}$ convergers to $ (1-m^{-s})^{-2}$

Question

I would please appreciate help showing: For $m$ a fixed integer

$\sum_{j=0}^\infty \frac{j+1}{m^{-js}}$ converges to $\frac{1}{(1-m^{-s})^2}$

There is a hint to treat the sum as a power series in $m^{-s}$.

A power series in $m^{-s}$ itself would converge to $\frac{m^s}{m^s-1}$ (for $m^s>1$).

Even if this is correct, that's as far as I could get, although the $(1-m^{-s})^{-2}$ looks like a derivative was taken.

Thanks

Answer 1

The identity $$ |x|<1\quad\Longrightarrow\quad \frac{1}{(1-x)^2}=\sum_{j\geq 0}(j+1)\,x^j $$ follows from stars and bars. Just set $x=\frac{1}{m^s}$.