Monday, 23 May 2016

calculus - Find formula of sum sin(nx)




I wonder if there is a way to calculate the



Sn=sinx+sin2x++sinnx




but using only derivatives ?


Answer



Using telescopic sums:



sin(mx)sin(x/2)=12(cos((m1/2)x)cos((m+1/2)x))
Hence:
Snsinx2=12(cosx2cos((n+12)x))=sinnx2sin(n+1)x2.


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