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Math Help - Infinite Product

  1. #1
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    Infinite Product

    1. Find the value of Π(from n=1 to infinity) (1 + 1/n^2).
    I know that I need to use the product expansion for sin
    πz but i do not know what to do


    2. Suppose that p is a polynomial of degree n and that |p(z)|
    ≤ M if |z| = 1. Show that |p(z)| ≥ M|z|^n if |z|≥ 1.


    I don't know how to solve this T-T


    Plz help me!!!

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  2. #2
    MHF Contributor chisigma's Avatar
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    The 'infinite product' of \frac{\sin x}{x} is...

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

    ... and from (1) we derive...

    \frac{\sin ix}{ix}= \frac{\sinh x}{x} = \prod_{n=1}^{\infty} (1+\frac{x^{2}}{\pi^{2}\cdot n^{2}}) (2)

    Setting in (2) x=\pi we obtain...

    \prod_{n=1}^{\infty} (1+\frac{1}{n^{2}}) = \frac{\sinh \pi}{\pi} (3)

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by dymin3 View Post
    2. Suppose that p is a polynomial of degree n and that |p(z)| ≤ M if |z| = 1. Show that |p(z)| ≥ M|z|^n if |z|≥ 1.
    If n>0 [i.e. the polynom ial is not a constant...], that is a particular case of Liouville's theorem...

    http://en.wikipedia.org/wiki/Liouvil...mplex_analysis)

    Kind regards

    \chi \sigma
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  4. #4
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    Thank you so much!!! Have a great day!!!
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