# Sum of Series of Product of Power and Sine

## 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 Corollary to Euler's Formula $\ds$ $=$ $\ds \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}$ after simplification

$\blacksquare$