Does there exist a closed form expression for the following sum:
Sum over n from 0 to infinity of:
Sin((2n+1)(pi)(a))/((pi)(2n+1))^(2k+3),
where a is a real number from 0 to 1, and k is an integer that is equal to or greater than 0.
I am looking for an answer only in terms of (a) and numerical constants.
I used Mathematica 12.3 to evaluate this sum, but the answer given was a rewrite of the sum.
If anyone knows of Tables of Sums or other sources where this sum is listed, please let me know.
Thanks very much for you kind help.
Dear Ron, there exist collections of tables of series (in this case, it might be regarded as a Fourier series), and you may try to look for them on Google. However, Mathematica usually contains nearly all the known closed-form results, and if Mathematica provides no answer, then the series is likely to have no known sum.
(note however that, due to the fast decrease of its terms, the series tends to coincide with the first term except for very low values of k)
It is probably not possible for this series to converge to a simple expression if a known function (this is unfortunately the general case, very few functions are named (trigonometric functions, logarithms, polynomials) but not the others.
Dear Ron,
Mathematica gives the answer, but I'm not sure that such form can be considered as closed.
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