Infinite Product.

• Aug 17th 2011, 11:18 PM
Also sprach Zarathustra
Infinite Product.
Something nice that came across with:

Is the product $\prod_{n=0}^{\infty}\frac{\beta +n}{\alpha+n }$ converge or diverge?
• Aug 17th 2011, 11:45 PM
CaptainBlack
Re: Infinite Product.
Quote:

Originally Posted by Also sprach Zarathustra
Something nice that came across with:

Is the product $\prod_{n=0}^{\infty}\frac{\beta +n}{\alpha+n }$ converge or diverge?

Nice. Obvious when $\alpha=\beta$, when these are not equal taking logs and considering the convergence of the resulting series seems to work a treat.

(If $\alpha$ and/or $\beta$ are/is negative a bit more care is needed but it still seems to work)

CB
• Aug 18th 2011, 12:10 AM
Also sprach Zarathustra
Re: Infinite Product.
Quote:

Originally Posted by CaptainBlack
Nice. Obvious when $\alpha=\beta$, when these are not equal taking logs and considering the convergence of the resulting series seems to work a treat.

CB

Yes! (Wondering)

Spoiler:
$\prod_{n=0}^{\infty}\frac{\beta +n}{\alpha+n } \equiv \prod_{n=0}^{\infty} \left( 1-\frac{\alpha -\beta }{\alpha +n} \right )$