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Math Help - convergence of product

  1. #1
    Senior Member Dinkydoe's Avatar
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    convergence of product

    Is the following product finite of infinite...?

    \prod_{k=1}^{\infty}\left(1+\frac{1}{k(k+2)}\right  )^{\frac{k}{\log 2}}

    Is there some convergence criterea i can use?
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  2. #2
    MHF Contributor
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    Re: convergence of product

    Hey Dinkydoe.

    You might want to consider if you can find a relationship with an exponential function (since this can be written in a limiting form similar to that) and show if you get a situation where only a finite number of terms are greater than 1 in absolute value.

    If you can show that for example after a certain point, that all values are less than one in magnitude then the the product will converge.

    You can also show that if the logarithm of the product converges then the product converges as well.
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  3. #3
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    Re: convergence of product

    The product \prod_{n=1}^\infty(1+a_n) converges if and only if the sum \sum_{n=1}^\infty{a_n} converges. And

    \left(1+\frac{1}{k(k+2)}\right )^{\frac{k}{\log 2}}\ge{1}+\left(\frac{k}{\log 2}\right)\left(\frac{1}{k(k+2)}\right)=1+\frac{1}{  (k+2)\log{2}}

    so it diverges.

    - Hollywood
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