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)=θ−θ33!+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 θ∈R such that sin(θ/2)≠0,
n∑k=0sin(kθ)=ℑn∑k=0eikθ=ℑ(ei(n+1)θ−1eiθ−1)=ℑ(ei(n+1)θ/2(ei(n+1)θ/2−e−i(n+1)θ/2)eiθ/2(eiθ/2−e−iθ/2))=ℑ(einθ/2(2isin((n+1)θ/2))(2isin(θ/2)))=ℑ(einθ/2sin((n+1)θ/2)sin(θ/2))=ℑ((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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