We can use also this well known summation formula (a consequence of the residue theorem): so if we take we get 𝕫 as wanted.
Beginner2022-01-23Added 13 answers
In this answer (using complex methods) and in this answer (using real methods), it is shown in detail that
(1) is the sum for even and odd k. The sum for even k would be
The sum for even minus the sum for odd would be twice (2) minus (1)
Skilled2022-01-27Added 437 answers
We begin by expanding the function in a Fourier series,
for . The Fourier coefficients (1) are given by
Substituting (2) into (1), setting , and dividing by reveals
Finally, enforcing the substitution and dividing by in (3) yields the coveted result
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