I want to compute that $\sum_{i=2}^{n}\dfrac{i^{m+1}-1}{i-1}+m=\sum_{j=0}^n\sum_{i=1}^m i^j$. If you closely notice I just rewrote summation of power sum. I tried to search google for any known identity, but I couldn't find anything. Another thing I want to confirm is that $\underbrace{\Pi_{a=1}^n\Pi_{b=1}^a\cdots\Pi_{r=1}^p}_{m \text{ times}}=\dfrac{(n+m-1)!}{n!m!}$. I don't have any proof for this but I found it by fixing some $m$ and running wolfram on this series. If you know any series identities which can simplify these, please let me know.
Any help is appreciated!