Sequence whose terms are sums of Egyptian fractions Let $M$ be a positive integer. Define a nonnegative sequence $(s_p)_{p\geq 2}$ by
$$
s_p=\sum_{1 \leq i \leq j \leq p} \frac{1}{(M+(j-i+1))\prod_{l=j+1}^{p}(M+l)}
$$
I conjecture that $s_p\to +\infty$ when $p\to\infty$. Can anyone prove
or disprove ?

We have:
$$s_p = \sum_{j=1}^{p}\sum_{1\leq i\leq j}\frac{1}{M+1+(j-i)}\frac{\Gamma(M+j+1)}{\Gamma(M+p+1)}$$
hence:
$$ s_p = \frac{1}{(M+p)!}\sum_{j=1}^{p}\left(H_{M+j}-H_{M}\right)(M+j)!$$
and by considering only the $j=p$ term we get:
$$ s_p > \sum_{h=1}^{p}\frac{1}{M+h}$$
where the RHS is diverging as $p\to+\infty$ by comparison with the harmonic series. 
The RHS behaves like $\log\left(1+\frac{p}{M}\right)$.
