Question

Prove that if x is not a negative integer , summation n=1 to infinity of 1/(x+n)(x+n+1) = 1/1+x .

Answer 1

$\sum^{\infty}_{n=1}\dfrac1{(x+n)(x+n+1)}=\sum^{\infty}_{n=1}(\dfrac1{x+n}-\dfrac1{x+n+1})=$

$=\lim_{N\rightarrow \infty}\sum^{N}_{n=1}(\dfrac1{x+n}-\dfrac1{x+n+1})= \lim_{N\rightarrow \infty}(\dfrac1{x+1}-\dfrac1{x+N+1})=$

$=\dfrac1{x+1}-\lim_{N\rightarrow \infty}\dfrac1{x+N+1}=\dfrac1{x+1}$