real analysis - How to derive the following identity?

The book Irresistible Integrals by George Boros and Victor Moll on page 204 has the following identity

$\displaystyle \frac{1}{1+x}=\prod_{k=1}^{\infty}\left(\frac{k+x+1}{k+x} \times \frac{k}{k+1}\right)$

How does one derive this?

Thanks.

Answer

Multiply out

$$\prod_{k=1}^N\left(\frac{k+x+1}{k+x} \times \frac{k}{k+1}\right)$$

and cancel like terms. You will be left with

$$\frac{1}{1+x}\left(\frac{N+x+1}{N+1}\right) .$$