Friday, 8 July 2016

calculus - Calculating sumnk=0sin(ktheta)




I'm given the task of calculating the sum ni=0sin(iθ).



So far, I've tried converting each sin(iθ) in the sum into its taylor series form to get:
sin(θ)=θθ33!+θ55!θ77!...
sin(2θ)=2θ(2θ)33!+(2θ)55!(2θ)77!...
sin(3θ)=3θ(3θ)33!+(3θ)55!(3θ)77!...
...
sin(nθ)=nθ(nθ)33!+(nθ)55!(nθ)77!...




Therefore the sum becomes,
θ(1+...+n)θ33!(13+...+n3)+θ55!(15+...+n5)θ77!(17+...+n7)...



But it's not immediately obvious what the next step should be.



I also considered expanding each sin(iθ) using the trigonemetry identity sin(A+B), however I don't see a general form for sin(iθ) to work with.


Answer



You may write, for any θR such that sin(θ/2)0,
nk=0sin(kθ)=nk=0eikθ=(ei(n+1)θ1eiθ1)=(ei(n+1)θ/2(ei(n+1)θ/2ei(n+1)θ/2)eiθ/2(eiθ/2eiθ/2))=(einθ/2(2isin((n+1)θ/2))(2isin(θ/2)))=(einθ/2sin((n+1)θ/2)sin(θ/2))=((cos(nθ/2)+isin(nθ/2))sin((n+1)θ/2)sin(θ/2))=sin(nθ/2)sin((n+1)θ/2)sin(θ/2).


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