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There exist positive integers A,B,C,D,E,FA,B,C,D,E,FA,B,C,D,E,F such that ∑p=1ncot6(pπ2n+1)=1Fn(2n−1)(An4+Bn3+Cn2−Dn+E) \sum_{p=1}^n \cot^6 \left(\tfrac{p \pi}{2n+1}\right) = \tfrac{1}{F}n(2n-1)\big(An^4 + Bn^3 + Cn^2 - Dn + E\big) p=1∑ncot6(2n+1pπ)=F1n(2n−1)(An4+Bn3+Cn2−Dn+E) for all positive integers nnn, where FFF is as small as possible.
What is A+B+C+D+E+F?A + B + C + D + E + F?A+B+C+D+E+F?
Bonus: Use this result to prove that ∑p=1∞1p6=1945π6. \sum_{p=1}^\infty \tfrac{1}{p^6} = \tfrac{1}{945}\pi^6. p=1∑∞p61=9451π6.
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