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Math Help - How to calculate the limit of this product?

  1. #1
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    How to calculate the limit of this product?

    \prod_{n=2}^{\infty}(1-\frac{1}{n^2})=\lim_{n\to\infty}(1-\frac{1}{4})(1-\frac{1}{9})\dots (1-\frac{1}{n^2})

    Thank you!
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: How to calculate the limit of this product?

    Hi gotmejerry!

    (1-{1\over n^2}) = {(n-1)(n+1) \over n^2}

    Can you write out a couple of factors and see how things cancel?
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  3. #3
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    Re: How to calculate the limit of this product?

    No unfortunately I cant.
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: How to calculate the limit of this product?

    Which factor do you get if you fill in n=2?
    And which factor for n=3?
    And n=4?
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  5. #5
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    Re: How to calculate the limit of this product?

    3/4 8/9 15/16 but what do i do with them, how do they cancel each other?
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: How to calculate the limit of this product?

    Let's write that out:

    {1 \cdot 3 \over 2^2} \cdot {2 \cdot 4 \over 3^2} \cdot {3 \cdot 5 \over 4^2} \cdot {4 \cdot 6 \over 5^2} \cdot ... \cdot {(n-1)  (n+1) \over n^2}

    Do you see numbers that cancel?
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  7. #7
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    Re: How to calculate the limit of this product?

    Yes, thanks, so If I am not wrong, only 1/2 *(n+1)/n remains? And its limit is 1/2. Thank you very much
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  8. #8
    Super Member ILikeSerena's Avatar
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    Re: How to calculate the limit of this product?

    Right!
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  9. #9
    MHF Contributor chisigma's Avatar
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    Re: How to calculate the limit of this product?

    Quote Originally Posted by gotmejerry View Post
    \prod_{n=2}^{\infty}(1-\frac{1}{n^2})=\lim_{n\to\infty}(1-\frac{1}{4})(1-\frac{1}{9})\dots (1-\frac{1}{n^2})

    Thank you!
    Remembering the 'infinite product'...

     \frac{\sin \pi x}{\pi x}= \prod_{n=1}^{\infty} (1- \frac{x^{2}}{n^{2}}) (1)

    ... Your 'infite product' seems to be [applying l'Hopital rule] ...

    \prod_{n=2}^{\infty} (1- \frac{1}{n^{2}}) = \lim_{x \rightarrow 1} \frac{\sin \pi x}{\pi x (1-x^{2})} = \frac{1}{2}(2)



    Marry Christmas from Serbia

    \chi \sigma
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