Sum of Infinite Series of Product of nth Power of Cosine by nth Multiple of Cosine

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Theorem

Let $0 < \theta < \dfrac \pi 2$.


Then:

\(\ds \sum_{n \mathop = 0}^\infty \cos^n \theta \, \map \cos {n + 1} \theta\) \(=\) \(\ds \cos \theta + \cos \theta \cos 2 \theta + \cos^2 \theta \cos 3 \theta + \cos^3 \theta \cos 4 \theta + \cdots\)
\(\ds \) \(=\) \(\ds 0\)


Proof

Let $0 < \theta < \dfrac \pi 2$.

Then $0 < \cos \theta < 1$.

\(\ds \sum_{k \mathop = 0}^\infty r^k \cos k \theta\) \(=\) \(\ds \dfrac {1 - r \cos \theta} {1 - 2 r \cos \theta + r^2}\) Sum of Infinite Series of Product of Power and Cosine: $\size r < 1$
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \cos^k \theta \cos k \theta\) \(=\) \(\ds \dfrac {1 - \cos^2 \theta } {1 - 2 \cos^2 \theta + \cos^2 \theta}\) setting $r = \cos \theta$
\(\ds \leadsto \ \ \) \(\ds \cos^0 \theta \cos 0 \theta + \sum_{k \mathop = 1}^\infty \cos^k \theta \cos k \theta\) \(=\) \(\ds \dfrac {1 - \cos^2 \theta } {1 - \cos^2 \theta}\) simplifying
\(\ds \leadsto \ \ \) \(\ds 1 + \sum_{k \mathop = 1}^\infty \cos^k \theta \cos k \theta\) \(=\) \(\ds \dfrac {1 - \cos^2 \theta } {1 - \cos^2 \theta}\) simplifying
\(\ds \leadsto \ \ \) \(\ds 1 + \sum_{k \mathop = 1}^\infty \cos^k \theta \cos k \theta\) \(=\) \(\ds 1\) simplifying
\(\ds \leadsto \ \ \) \(\ds \cos \theta \sum_{k \mathop = 1}^\infty \cos^{k - 1} \theta \, \map \cos {n + 1} \theta\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 1}^\infty \cos^{k - 1} \theta \cos k \theta\) \(=\) \(\ds 0\) dividing by $\cos \theta$
\(\ds \leadsto \ \ \) \(\ds \sum_{k \mathop = 0}^\infty \cos^k \theta \cos k \theta\) \(=\) \(\ds 0\) Translation of Index Variable of Summation

$\blacksquare$


Sources

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