Summations of series double et produit double

Question

I need help with these summations if you can provide me with detailed answers it would be great:

$S_{n}=\sum_{k=1}^{n}\left(1+\frac{1}{k^2}+\frac{1}{(1+k)^2}\right)^{1/2}$

$S_{n}=\sum_{k=1}^{n} (k-1)^2k!$

$\prod_{0<i<j<n} ij $

Answer 1

Hints only.

a) Compute $\displaystyle (1+\frac{1}{k}-\frac{1}{k+1})^2$.

b) Show that $\displaystyle (k-1)^2=(k+1)(k+2)-5(k+1)+4$.

c) Let $\displaystyle P=\prod_{1\leq i<j\leq n-1} ij$, then $\displaystyle P=\prod_{1\leq j<i\leq n-1} ij$, and $$\prod_{1\leq i\leq n-1,1\leq j\leq n-1} ij=(\prod_{i=1}^{n-1} i^2)(\prod_{1\leq i<j\leq n-1} ij)(\prod_{1\leq j<i\leq n-1} ij)$$