Tuesday, September 24, 2019

Sum of series of the form sumfrac(1)m+12m+1sin(fracpi2(2m+1)x)

I would like to calculate the sum of the series:

m=M+1(1)m+12m+1sin((2m+1)π2x)
where M is big and finite.






I searched on the books and found this sum:
k=1(1)k12k1sin((2k1)x)=12lntan(π4+x2)
Now I try to put my question in the above form. Let m-M=n and m=n+M
n=1(1)n+M+12(n+M)+1sin(2(n+M)+1)π2x)
I do not know how to proceed further.

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