What is wrong with this series?

Calculus Level 5

∑ n = 1 , 3 , 5 , … ∞ k sin ⁡ 2 ( n k ) n 2 \large \sum_{n=1,3,5,\ldots}^\infty \dfrac{k \sin^2 \left( \frac nk \right) } {n^2}

Let k k be a positive number that is not a multiple of 1 π \dfrac1\pi .

The series above equals to A B π C \dfrac AB \pi^C for positive integers A , B A,B and C C with gcd ⁡ ( A , B ) = 1 \gcd(A,B) = 1 , find the value of A + B + C A+B+C .


The answer is 6.

Alisa Meier by Alisa Meier , 24, Germany

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1 solution

∑ n = 0 ∞ k sin ⁡ 2 ( 2 n + 1 k ) ( 2 n + 1 ) 2 ⟹ − 1 16 e − 2 i k k ( Φ ( e − 4 i k , 2 , 1 2 ) + e 4 i k Φ ( e 4 i k , 2 , 1 2 ) − π 2 e 2 i k ) \sum _{n=0}^{\infty } \frac{k \sin ^2\left(\frac{2 n+1}{k}\right)}{(2 n+1)^2} \Longrightarrow\\ -\frac{1}{16} e^{-\frac{2 i}{k}} k \left(\Phi \left(e^{-\frac{4 i}{k}},2,\frac{1}{2}\right)+e^{\frac{4 i}{k}} \Phi \left(e^{\frac{4 i}{k}},2,\frac{1}{2}\right)-\pi ^2 e^{\frac{2 i}{k}}\right) Rationalize [ Table [ Chop [ N [ t π ] ] , { k , Range [ 100 ] } ] ] ⟹ a table of 1 4 . \text{Rationalize}\left[\text{Table}\left[\text{Chop}\left[N\left[\frac{t}{\pi }\right]\right],\{k,\text{Range}[100]\}\right]\right] \Longrightarrow \text{a table of} \frac14.

A=1, b=4 and C=1 for a total of 6.

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