Saturday, 27 January 2018

trigonometry - Finding a closed form for cosx+cos3x+cos5x+cdots+cos(2n1)x




We have to find




g(x)=cosx+cos3x+cos5x++cos(2n1)x




I could not get any good idea .




Intialy I thought of using



cosa+cosb=2cos(a+b)/2cos(ab)/2


Answer



Let z=cosθ+isinθ i.e. z=eiθ



Your sum:eiθ+e3iθ+e5iθ+...e(2n1)iθ



This is a GP with common ratio e2iθ




Therefore sum is a(rn1)r1


eiθ(e2niθ1)e2iθ1

(cosθ+isinθ)(cos(2nθ)+isinθ1)cos(2θ)+isin(2θ)1



Computing it's real part should give you the answer



Acknowledgement:Due credits to @LordShark Idea


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...