Sunday, March 10, 2019

complex numbers - Solve sumnr=1(2r1)cos(2r1)theta



How should i solve : nr=1(2r1)cos(2r1)θ



I can solve nr=1cos(2r1)θ by considering
nr=1z2r1 and using summation of geometric series, but I can't seem to find a common geometric ratio when 2r1 is involved in the summation.




Visually : nr=1z2r1=z+z3+...+z2r1 where the common ratio r=z2 can easily be seen, but in the case of nr=1(2r1)z2r1=z+3z3+5z5+...+(2r1)z2r1, how should i solve this ? A hint would be appreciated.


Answer



Note that nr=1(2r1)cos(2r1)θ=nr=1ddθsin(2r1)θ=ddθnr=1sin(2r1)θ
Now calculate nr=1sin(2r1)θ=sin2(nθ)sin(θ) Through the formula for the sum of sin's.



Now just diffferentiate with regard to θ.


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