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Let $u_n = \dfrac{1}{n}\sum_{i=1}^{n}\left(\dfrac{i}{n}\right)^2\,\sin\left(\dfrac{i\,\pi}{n}\right)$.

Prove that $$\lim\limits_{n\rightarrow \infty}\dfrac{1}{n}\sum_{i=1}^{n}\left(\dfrac{i}{n}\right)^2\,\sin\left(\dfrac{i\,\pi}{n}\right) = \dfrac{1}{\pi} - \dfrac{4}{\pi^3}$$

I know that I can express $\sin\left(\dfrac{i \pi}{n}\right)$ with complex, using Euler's relation $\sin \theta = \dfrac{1}{2i}(e^{i\theta}-e^{-i\theta})$, but I don't know how to use (or even if I can use it).

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"Prove" is a verb, as in "Prove that...". "Proof" is a noun, as in "Provide me with a proof that..." –  Pedro Tamaroff Mar 18 '13 at 18:45
    
Your notation is a bit confusing. $i$ as iterator under the sum sign and as $\sqrt{-1}$ confused (at least) me. –  dot dot Mar 18 '13 at 18:53
    
@dotdot It is common to use $i$ when indexing a sum. You will also see $j,k$, for example. –  Pedro Tamaroff Mar 18 '13 at 18:56
    
@PeterTamaroff: I normally use $m$,$n$ or $q$ :). –  dot dot Mar 18 '13 at 18:57
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1 Answer

up vote 6 down vote accepted

This is a Riemann sum:

$$\lim_{n \rightarrow \infty}\dfrac{1}{n}\sum_{i=1}^{n}\left(\dfrac{i}{n}\right)^2\,\sin\left(\dfrac{i\,\pi}{n}\right) = \int_0^1 dx \: x^2 \sin{\pi x}$$

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You missed $\pi$! –  Pedro Tamaroff Mar 18 '13 at 18:46
    
@PeterTamaroff: thanks. –  Ron Gordon Mar 18 '13 at 18:47
    
You also beat me to it. Why did I waste time with grammar? =) –  Pedro Tamaroff Mar 18 '13 at 18:48
    
....and you integrate by parts. –  Sami Ben Romdhane Mar 18 '13 at 18:48
    
@SamiBenRomdhane Well, let's leave something for the OP to work out. –  Pedro Tamaroff Mar 18 '13 at 18:49
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