Gamma Function- Limit/ Product Representation-

Hello,

I am trying to figure out how the product representation of the Gamma function can be rewritten as this limit.

I've tried working on it for hours and I still cannot figure it out.

First I worked trying to change the product representation into the limit, and the other way around and still couldn't figure it out.

The limit representation is given by:

$\displaystyle \Gamma(z)=\lim_{n\rightarrow\infty}\frac{n!n^z}{z( z+1)(z+2)...(z+n)}$

The product representation is given by:

$\displaystyle \Gamma(z)=\frac{1}{z}\prod^\infty_{n=1}\frac{(1+\f rac{1}{n})^z}{1+\frac{z}{n}}$

I understand how the denominators are equal after some manipulation, but I can't quite understand the numerators.

Trying to manipulate the limit definition, this is what I've done:

$\displaystyle \Gamma(z)=\frac{1}{z}\lim_{n\rightarrow\infty}\fra c{n!n^z}{(z+1)(z+2)...(z+n)}$

I then just focused on changing the denominator of the limit into an infinite product.

$\displaystyle \frac{1}{z}\lim_{n\rightarrow\infty}\frac{1}{(z+1) (z+2)...(z+n)}=\frac{1}{z}\prod^\infty_{n=1}\frac{ 1}{z+n}$

$\displaystyle \frac{1}{z}\prod^\infty_{n=1}\frac{1}{z+n}\frac{\f rac{1}{n}}{\frac{1}{n}}=\frac{1}{z}\prod^\infty_{n =1}\frac{n^{-1}}{(1+\frac{z}{n})}$

So I see how the denominators resemble each other. However, I can't quite figure out the numerators.

I've done some more manipulations with the numerator which I won't show here out of tediousness, but the work didn't get me anywhere.

It would be greatly appreciated if an explanation was posted on how to convert the given infinite product definition into the limit definition, or the limit definition into the infinite product.

Thank you.