Sum of Series of Product of Power and Sine
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Theorem
Let $r \in \R$ such that $\size r < 1$.
Then:
\(\ds \sum_{k \mathop = 1}^n r^k \map \sin {k x}\) | \(=\) | \(\ds r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots + r^n \sin n x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}\) |
Proof
From Euler's Formula:
- $e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
\(\ds \sum_{k \mathop = 1}^n r^k \map \sin {k x}\) | \(=\) | \(\ds \map \Im {\sum_{k \mathop = 1}^n r^k e^{i k x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\sum_{k \mathop = 0}^n \paren {r e^{i x} }^n}\) | as $\map \Im {e^{i \times 0 \times x} } = \map \Im 1 = 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {1 - r^{n + 1} e^{i \paren {n + 1} x} } {1 - r e^{i x} } }\) | Sum of Infinite Geometric Sequence: valid because $\size r < 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {\paren {1 - r^{n + 1} e^{i \paren {n + 1} x} } \paren {1 - r e^{-i x} } } {\paren {1 - r e^{-i x} } \paren {1 - r e^{i x} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {r^{n + 2} e^{i n x} - r^{n + 1} e^{i \paren {n + 1} x} - r e^{-i x} + 1} {1 - r \paren {e^{i x} + e^{- i x} } + r^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\frac {r^{n + 2} \paren {\cos n x + i \sin n x} - r^{n + 1} \paren {\map \cos {n + 1} x + i \map \sin {n + 1} x} - r \paren {\cos x - i \sin x} + 1} {1 - 2 r \cos x + a^2} }\) | Euler's Formula and Euler's Formula: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}\) | after simplification |
$\blacksquare$
Also see
- Sum of Infinite Series of Product of Power and Sine
- Sum of Infinite Series of Product of Power and Cosine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Miscellaneous Series: $19.44$