Seeking proofs of a limit expression of the gamma function

Question

I just started reading H. M. Edwards' Riemann's Zeta Function and I encountered an equation that I don't know how to prove. It's Equation (3) on page 8: $$\Pi(s)=\lim_{N\to\infty}\frac{1·2\cdots N}{(s+1)(s+2)\cdots(s+N)}(N+1)^s.$$ Both rigorous proofs and informal intuitions are welcome. The definition of the gamma function is given by Equation (2) on the same page: $$\Pi(s)=\int_0^\infty e^{-x}x^sdx\quad(\Re s>-1).$$

Answer 1

Elaborating what's suggested in my comment above (somehow this lacks a proof)...

Consider $I_n(s)=\displaystyle\int_0^n x^s\left(1-\frac{x}{n}\right)^n\,dx$. Integrating by parts $n$ times, we see that $$I_n(s)=\frac{n!}{n^n(1+s)\ldots(n+s)}\int_0^n x^{n+s}\,dx=\frac{n^{s+1}\cdot n!}{(1+s)\ldots(n+1+s)}.$$ Now we're going to apply DCT with $g(x)=x^s e^{-x}$ and $$f_n(x)=\begin{cases}x^s \displaystyle\left(1-\frac{x}{n}\right)^n,& 0<x<n\\ \hfill 0,\hfill &\hfill x\geqslant n\hfill\end{cases}$$ which is valid because $f_n(x)\leqslant g(x)$ follows from $1-t\leqslant e^{-t}$.

This gives $\displaystyle\lim_{n\to\infty}I_n(s)=\Pi(s)$. Your limit $\displaystyle\lim_{n\to\infty}I_n(s)\frac{(n+1)^s(n+s+1)}{n^{s+1}}$ is the same.