Sum of Infinite Series of Product of Power and Cosine
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Theorem
Let $r \in \R$ such that $\size r < 1$.
Then:
\(\ds \sum_{k \mathop = 0}^\infty r^k \cos k x\) | \(=\) | \(\ds 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - r \cos x} {1 - 2 r \cos x + r^2}\) |
Proof
From Euler's Formula:
- $e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
\(\ds \sum_{k \mathop = 0}^\infty r^k \cos k x\) | \(=\) | \(\ds \map \Re {\sum_{k \mathop = 0}^\infty r^k e^{i k x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\sum_{k \mathop = 0}^\infty \paren {r e^{i x} }^k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac 1 {1 - r e^{i x} } }\) | Sum of Infinite Geometric Sequence: valid because $\size r < 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {1 - r e^{-i x} } {\paren {1 - r e^{-i x} } \paren {1 - r e^{i x} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {1 - r e^{-i x} } {1 - r \paren {e^{i x} + e^{- i x} } + r^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\frac {1 - r \paren {\cos x - i \sin x} } {1 - 2 r \cos x + r^2} }\) | Euler's Formula: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - r \cos x} {1 - 2 r \cos x + r^2}\) | after simplification |
$\blacksquare$
Also see
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.4$. Power Series
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Miscellaneous Series: $19.41$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.): $2$: Miscellaneous Problems: $46 \ \text{(a)}$