prove algebraicly the sum of the reciprocals of 2 consecutive odd numbers always produces the leg of a pythagorem triple

$\displaystyle 1/3+1/5=8/15$

$\displaystyle 8^2+15^2=17^2$

let $\displaystyle 2n-1$ and $\displaystyle 2n+1$ be the two odd natural numbers..

$\displaystyle 1/(2n-1)+1/(2n+1)=(2n-1)+(2n+1)/(2n-1)(2n+1)$

$\displaystyle =4n/4n^2 -4n+1$

thats as fathest as i can get