Thread: Leibniz Limits... View Single Post
 HW Helper P: 1,025 A problem: From the following i. $$\lim_{q\rightarrow\infty}\prod_{n=p}^{pq} \left(1+\frac{x}{n}\right)=p^{x},$$ and ii.$$\lim_{n\rightarrow\infty}\left(1-\frac{t}{n}\right)^{n}=e^{-t},$$ reason that $$\int_{t=0}^{\infty}e^{-t}t^{x}dt=\lim_{n\rightarrow\infty}\frac{1\cdot 2\cdots n}{(1+x)(2+x)\cdots (n+x)}n^{x},$$ where x is complex and not a negative integer. I quoted this exercise from Introduction to the Theory of Analytic Functions by Harkness & Morley that I happed to have a single page of printed (pg. 208), (i) is the from a seperate exercise listed immediately prior to the exercise at hand in which (ii) is a given.