Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}$ How to prove the following equality?
$$\sum_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$

Another approach without using hidden beta and gamma functions where one gets to excercise manipulation of binomial coefficients and generating functions is to use
$$\prod_{k=1}^m \frac{1}{n+k} =
\frac{1}{(m-1)!} \sum_{k=0}^{m-1} (-1)^k \binom{m-1}{k} \frac{1}{n+k+1}.$$
Now compute how often the fraction $1/q$ where $q\ge m+1$ occurs on the right side and with what coefficients. It occurrs for the first time in the term for $n=q-m$ and for the last time in $n=q-1,$ so its coefficient is (keeping in mind that $k = q-n-1$)
$$\sum_{n=q-m}^{q-1} (-1)^{q-n-1} \binom{m-1}{q-n-1}.$$
Put $p=q-n-1$ in the above sum to get
$$\sum_{p=q-(q-m)-1}^{q-(q-1)-1} (-1)^p \binom{m-1}{p} =
\sum_{p=m-1}^0 (-1)^p \binom{m-1}{p} = \sum_{p=0}^{m-1} (-1)^p \binom{m-1}{p} = 0,$$
so for $q\ge m+1$ the sum telescopes and the contribution of the value $1/q$ is zero.
The leftover contribution is
$$\sum_{q=2}^m \frac{1}{q}\sum_{n=1}^{q-1} (-1)^{q-n-1}  \binom{m-1}{q-n-1}
= \sum_{q=2}^m \frac{1}{q}\sum_{n=0}^{q-2} (-1)^n \binom{m-1}{n}$$
To conclude we ask where $\binom{m-1}{n}$ appears and for what values of $q$, getting
$$ \sum_{n=0}^{m-2}  (-1)^n \binom{m-1}{n} \sum_{q=2+n}^m \frac{1}{q}
= \sum_{n=0}^{m-2}  (-1)^n \binom{m-1}{n} \left(H_m - H_{n+1}\right).$$
Split the sum in two to obtain first,
$$H_m \sum_{n=0}^{m-2}  (-1)^n \binom{m-1}{n} =
H_m (0 - (-1)^{m-1} ) = (-1)^m H_m$$
and second,
$$ \sum_{n=0}^{m-2}  (-1)^n \binom{m-1}{n} H_{n+1} =
\sum_{n=0}^{m-1}  (-1)^n \binom{m-1}{n} H_{n+1} - (-1)^{m-1} H_m =
\sum_{n=0}^{m-1}  (-1)^n \binom{m-1}{n} H_{n+1} + (-1)^m H_m$$
so the difference between the two is
$$- \sum_{n=0}^{m-1}  (-1)^n \binom{m-1}{n} H_{n+1}.$$
This is the binomial transform of $H_{n+1}$ and since
$$\sum_{n\ge 1} H_{n+1} z^n = f(z) = \frac{1}{z} \frac{1}{1-z} \log \frac{1}{1-z}$$
the sum has generating function
$$\frac{1}{1-z} f\left(\frac{z}{z-1}\right) = 
\frac{1}{z} (1-z) \log \frac{1}{1-z} = \left(\frac{1}{z}-1\right) \log \frac{1}{1-z}
\\= \sum_{m\ge 1} \left(\frac{1}{m+1}-\frac{1}{m}\right) z^m
= -\sum_{m\ge 1} \frac{1}{m(m+1)} z^m.$$
Observe that we need the coefficient of $z^{m-1}.$ It follows that the original sum has value
$$ \frac{1}{(m-1)!}  \frac{1}{(m-1)m} = \frac{1}{(m-1)m!}$$
as was to be shown.
I will prove the partial fraction identity in an additional answer.
