Question 2.46: Prove that if |a| < 1, (a) 1 + a cos θ + a^2 cos 2θ + a^3...
Prove that if |a| < 1,
(a) 1+a \cos \theta+a^2 \cos 2 \theta+a^3 \cos 3 \theta+\cdots=\frac{1-a \cos \theta}{1-2 a \cos \theta+a^2}
(b) a \sin \theta+a^2 \sin 2 \theta+a^3 \sin 3 \theta+\cdots=\frac{a \sin \theta}{1-2 a \cos \theta+a^2}
Let z=a e^{i \theta} in Problem 2.41. We can do this since |z|=|a|<1. Then
1+a e^{i \theta}+a^2 e^{2 i \theta}+a^3 e^{3 i \theta}+\cdots=\frac{1}{1-a e^{i \theta}}or
\begin{aligned} \left(1+a \cos \theta+a^2 \cos 2 \theta+\cdots\right)+i\left(a \sin \theta+a^2 \sin 2 \theta+\cdots\right) &=\frac{1}{1-a e^{i \theta}} \cdot \frac{1-a e^{-i \theta}}{1-a e^{-i \theta}} \\ &=\frac{1-a \cos \theta+i a \sin \theta}{1-2 a \cos \theta+a^2} \end{aligned}The required results follow on equating real and imaginary parts.
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