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Question-proof-n-sin-n-n-2-pi-nice-calculus-




Question Number 139404 by mnjuly1970 last updated on 26/Apr/21
          # Question ::             proof :: Σ_(n=−∞) ^( ∞) ( ((sin(n))/n) )^2 =π               ....... nice .calculus ......
$$\:\:\:\:\: \\ $$$$\:\:\:#\:{Question}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{proof}\:::\:\underset{{n}=−\infty} {\overset{\:\infty} {\sum}}\left(\:\frac{{sin}\left({n}\right)}{{n}}\:\right)^{\mathrm{2}} =\pi \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:…….\:{nice}\:.{calculus}\:……\:\:\:\:\: \\ $$
Answered by Dwaipayan Shikari last updated on 26/Apr/21
Σ_(n=−∞) ^∞ ((sin^2 (n))/n^2 )=2((π−1)/2)+1      Σ_(n=1) ^∞ ((sin^2 (nθ))/n^2 )=((θ(π−θ))/2)  =π
$$\underset{{n}=−\infty} {\overset{\infty} {\sum}}\frac{{sin}^{\mathrm{2}} \left({n}\right)}{{n}^{\mathrm{2}} }=\mathrm{2}\frac{\pi−\mathrm{1}}{\mathrm{2}}+\mathrm{1}\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}^{\mathrm{2}} \left({n}\theta\right)}{{n}^{\mathrm{2}} }=\frac{\theta\left(\pi−\theta\right)}{\mathrm{2}} \\ $$$$=\pi \\ $$
Commented by mnjuly1970 last updated on 26/Apr/21
thanks alot..
$${thanks}\:{alot}.. \\ $$

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