Tuesday, April 5, 2016

find sum of n terms of series sumcos(ntheta)


Use the result 1+z+z2...+zn=zn+11z1 to sum the series to n terms



1+cosθ+cos2θ+...


also show that partial sums of series cos(nθ) is bounded when 0<θ<π/2


My attempt


so z can be written as eiθ which means:


1+cosθ+cos2θ....+cosnθ+i(sinθ+sin2θ+....+sinnθ)=zn+11z1


after this.. i dont know


Answer



Remember that eit=cost+isinttC and that nj=0zj=1zn+11zzC,|z|<1. Thus nj=0cos(jθ)=nj=0(eijθ)=(nj=0(eijθ))=(1eiθ(n+1)1eiθ) The last term I wrote can be handled easily in order to be written explicitly and get the results you wanted.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...