I am learning the Gamma function, based on some lecture notes, and I wish to ask a couple of questions regarding infinite products.
Let $z$ be a complex number except $\{0, -1, \ldots \}$.
(1) How do I show that $\prod_{m=1}^n\left(1+\frac{z}{m}\right)^{-1}$ diverges as $n\rightarrow\infty$; but
(2) $\prod_{m=1}^n\left(1+\frac{z}{m}\right)^{-1}\left(1+\frac{1}{m}\right)^z$ converges?
Any suggested readings on the "limit-definition" of the Gamma function $\Gamma(z)$? Thanks!