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for-0-lt-r-1-and-x-R-2-find-S-n-0-r-n-cos-n-

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Question Number 30601 by abdo imad last updated on 23/Feb/18
for 0<r≤1 and (θ,x)∈R^2 find S=Σ_(n=0) ^∞ r^n cos(nθ).
for0<r⩽1and(θ,x)∈R2findS=∑n=0∞rncos(nθ).
Commented by abdo imad last updated on 24/Feb/18
S=Re(Σ_(n=0) ^∞ r^n e^(inθ) )but Σ_(n=0) ^∞ r^n e^(inθ) = Σ_(n=0) ^∞ (re^(iθ) )^n =(1/(1−re^(iθ) )) = (1/(1−r cosθ −irsinθ))=((1−r cosθ +isinθ)/((1−r cosθ)^2 +r^2 sin^2 θ)) ⇒ S= ((1−r cosθ)/((1−rcosθ)^2 +r^2 sin^2 θ))=((1−r cosθ)/(1−2rcosθ +r^2 )) but we must study the case r=1.
S=Re(∑n=0∞rneinθ)but∑n=0∞rneinθ=∑n=0∞(reiθ)n=11−reiθ=11−rcosθ−irsinθ=1−rcosθ+isinθ(1−rcosθ)2+r2sin2θ⇒S=1−rcosθ(1−rcosθ)2+r2sin2θ=1−rcosθ1−2rcosθ+r2butwemuststudythecaser=1.

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