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Math Help - Infinite Product.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Infinite Product.

    Something nice that came across with:


    Is the product \prod_{n=0}^{\infty}\frac{\beta +n}{\alpha+n } converge or diverge?
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  2. #2
    Grand Panjandrum
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    Re: Infinite Product.

    Quote Originally Posted by Also sprach Zarathustra View Post
    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
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Infinite Product.

    Quote Originally Posted by CaptainBlack View Post
    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!


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