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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series