Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can I rigorously and directly prove that $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$

share|cite|improve this question
2  
How do you define $\Gamma(z)$? – Pedro Tamaroff Aug 11 '13 at 1:23
up vote 6 down vote accepted

For simplicity define $z=k$ ,so we prove

$$\lim_{n\to \infty}\frac{n! \,\, n^k}{k(k+1)\cdots (k+n)}=(k-1)!$$

Note that

$$\frac{(k+n)!}{(k-1)!} = k(k+1)\cdots (k+n)$$

so we have

$$ (k-1)! \lim_{n\to \infty}\frac{ \, n! \,\, n^k}{(k+n)!}=(k-1)!\lim_{n\to \infty}\frac{n^k}{(k+n)(k+n-1)\cdots (n+1)}$$

Now this can be written as

$$(k-1)! \lim_{n\to \infty} \frac{n\cdot n\cdot \cdots n}{(n+k)(n+k-1)\cdots (n+1)}=(k-1)!$$

The gamma function is an extension of the factorial by the relation

$$\Gamma(k)=(k-1)!$$

share|cite|improve this answer
    
(+1) nice answer. – Mhenni Benghorbal Aug 11 '13 at 11:20
    
Thanks Mhenni :) – Zaid Alyafeai Aug 11 '13 at 20:51

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.