Question: Convert Fourier Series to Sum of Odd/Even Terms

Is there a way to convert a Fourier series (from the OrthogonalExpansions package) automatically into the sum of odd/even terms if the even/odd terms are 0 respectively?


 

restart

with(OrthogonalExpansions)

FourierSeries(x-2*Pi, x = 0 .. 4*Pi, infinity, series = cosine); simplify(subs(i = n, %))

Sum(8*((-1)^n-1)*cos((1/4)*n*x)/(n^2*Pi), n = 1 .. infinity)

(1)

Sum((-16*cos((1/4)*(2*k-1)*x))*(1/((2*k-1)^2*Pi)), k = 1 .. infinity)

Sum(-16*cos((1/4)*(2*k-1)*x)/((2*k-1)^2*Pi), k = 1 .. infinity)

(2)

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