3

I'm working on a problem and need this to converge to any value:

$\displaystyle \frac{(1/2)((1/2)-1)((1/2)-2)\cdots ((1/2)-n+1)}{(n+1)!} = \Pi_{j=1}^{n} \frac{\frac{1}{2} - j+1}{j+1}$

The convergence is evident when you begin doing the product. However, if there a way to prove this?

By the way, I do not know pi notation for products yet.

Hans Engler
  • 15,439
Grizzly0111
  • 509
  • \Pi_{i=m}^n a_i gives $\Pi_{i=m}^n a_i$ – Claude Leibovici Mar 20 '15 at 15:44
  • Do you know the gamma function ? – Claude Leibovici Mar 20 '15 at 15:47
  • @vadim123 Indeed, I noticed I had done a mistake in the transcription (since I don't know mathjax, I had copied an online template). I just fixed it. – Grizzly0111 Mar 20 '15 at 15:49
  • This is related to the binomial coefficient $\binom{1/2}{n}$. – Hans Engler Mar 20 '15 at 15:52
  • @ClaudeLeibovici unfortunately, I do not. This is part of a bonus question on my homework, so my prof must expect us to find things ourselves. – Grizzly0111 Mar 20 '15 at 15:56
  • If I am reading this correctly, there are $n$ factors in the numerator and $n+1$ factors in the denominator. For each term in the numerator, find a factor in the denominator that is just about the same size or maybe a little larger. This part of the product would therefore not become very large. Then look at the one term that is remaining in the denominator. What does this do to the product? The answer should now pop out at you. – Hans Engler Mar 20 '15 at 15:59
  • In any manner, I was not reading your expression correctly. – Claude Leibovici Mar 20 '15 at 16:03

0 Answers0