Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. We set
$$J _ { r , s } = \int _ { 0 } ^ { 1 } f _ { r , s } ( y ) \mathrm { d } y = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
Show that
$$J _ { r , s } = \sum _ { k = 0 } ^ { + \infty } \frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) }$$