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Question Number 132162 by Dwaipayan Shikari last updated on 11/Feb/21

Σ_(n=1) ^∞ ((sin(n))/n^2 )

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}\right)}{{n}^{\mathrm{2}} } \\ $$

Answered by mnjuly1970 last updated on 11/Feb/21

(1/(2i))[li_2 (e^i )−li_2 (e^(−i) )]

$$\frac{\mathrm{1}}{\mathrm{2}{i}}\left[{li}_{\mathrm{2}} \left({e}^{{i}} \right)−{li}_{\mathrm{2}} \left({e}^{−{i}} \right)\right] \\ $$

Commented by Dwaipayan Shikari last updated on 11/Feb/21

But exact result possible sir?

$${But}\:{exact}\:{result}\:{possible}\:{sir}? \\ $$

Commented by mnjuly1970 last updated on 11/Feb/21

i will try sir payan Σ((cos(n))/n^2 ) is very simple . Σ((sin(n))/n^2 ) is a little bit challanging fourier series is useful..

$$\:{i}\:{will}\:{try}\:{sir}\:{payan} \\ $$$$\:\Sigma\frac{{cos}\left({n}\right)}{{n}^{\mathrm{2}} }\:{is}\:{very}\:{simple}\:. \\ $$$$\Sigma\frac{{sin}\left({n}\right)}{{n}^{\mathrm{2}} }\:{is}\:{a}\:{little}\:{bit}\:{challanging} \\ $$$$\:{fourier}\:{series}\:{is}\:{useful}.. \\ $$

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