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Math Help - This is a problem under Riemann integral. Please help how to proceed.

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    This is a problem under Riemann integral. Please help how to proceed.

    lim n-infinity (1/n)(sin pi/n +sin 2pi/n + sin 3pi/n +...+sin npi/n) ?
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    MHF Contributor MarkFL's Avatar
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    Re: This is a problem under Riemann integral. Please help how to proceed.

    Recall:

    \int_a^b f(x)\,dx=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^nf\left(a+k\frac{b-a}{n} \right) \right]

    We are asked to evaluate:

    \lim_{n\to\infty}\left[\frac{1}{n}\sum_{k=1}^n\sin\left(\frac{k\pi}{n} \right) \right]

    So, let:

    a=0,b=1,f(x)=\sin(\pi x)

    Can you finish?
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    Senior Member MaxJasper's Avatar
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    Lightbulb Re: This is a problem under Riemann integral. Please help how to proceed.

    Quote Originally Posted by anushua View Post
    lim n-infinity (1/n)(sin pi/n +sin 2pi/n + sin 3pi/n +...+sin npi/n) ?
    Attached Thumbnails Attached Thumbnails This is a problem under Riemann integral. Please help how to proceed.-infinite-series.png  
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