Find the closed form of \(\displaystyle \displaystyle \sum_{r=1}^{n} \dfrac{1-\cos\left(\frac{(2r-1)\pi}{2n}\right)} {\left[1+\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]\left[5+3\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]^2} \)

r=1n1cos((2r1)π2n)[1+cos((2r1)π2n)][5+3cos((2r1)π2n)]2=n(8n11)16\large \sum_{r=1}^{n} \dfrac{1-\cos\left(\frac{(2r-1)\pi}{2n}\right)} {\left[1+\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]\left[5+3\cos\left(\frac{(2r-1)\pi}{2n}\right)\right]^2} = \dfrac{n(8n-11)}{16}

Prove that for integer n>1n>1, the equation above holds true.

Please try not to use induction (assuming it's possible).


This problem was copied from another math forum but no one responded to it.


This is a part of the set Formidable Series and Integrals.

#Calculus

Note by Aditya Kumar
5 years, 4 months ago

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@Aditya Kumar bache ki jaan lo ge kya thoda to hint de the Bhaiya.

Department 8 - 5 years, 4 months ago

have you tried roots of unity

Aareyan Manzoor - 5 years, 4 months ago

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How does that help? We can only use that if we can split out all the expressions and calculate them separately. fghf1g1h \sum \dfrac f{g \cdot h} \ne \sum f \cdot \sum \dfrac1g \cdot \sum \dfrac1h .

Pi Han Goh - 5 years, 4 months ago

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I havent tried this yet but we could use partial fractions and then f'/f or maybe chebyshev polynomials of the first kind.....

Aareyan Manzoor - 5 years, 4 months ago
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