Difference between revisions of "Taylor series of sine"

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==Theorem==
<strong>[[Taylor series of sine|Theorem]]:</strong> Let $z_0 \in \mathbb{C}$. The following [[Taylor series]] holds:
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Let $z_0 \in \mathbb{C}$. The following [[Taylor series]] holds:
$$\sin(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k(z-z_0)^{2k+1}}{(2k+1)!},$$
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$$\sin(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k z^{2k+1}}{(2k+1)!},$$
 
where $\sin$ denotes the [[sine]] function.
 
where $\sin$ denotes the [[sine]] function.
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<strong>Proof:</strong> █  
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==Proof==
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Using the [[Taylor series of the exponential function]] and the definition of $\sin$,
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$$\begin{array}{ll}
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\sin(z) &= \dfrac{e^{iz}-e^{-iz}}{2i} \\
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&= \dfrac{1}{2i} \left[ \displaystyle\sum_{k=0}^{\infty} \dfrac{i^k (z-z_0)^k}{k!} - \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k i^k (z-z_0)^k}{k!} \right] \\
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&= \dfrac{1}{2i} \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k).
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\end{array}$$
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Note that if $k=2n$ is a positive even integer, then
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$$i^k(1-(-1)^k)=i^{2n}(1-(-1)^{2n})=0,$$
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and if $k=2n+1$ is a positive odd integer, then
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$$i^k(1-(-1)^k)=i^{2n+1}(1-(-1)^{2n+1})=2i(-1)^n.$$
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Hence we have derived
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$$\begin{array}{ll}
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\sin(z)&=\dfrac{1}{2i} \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k) \\
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&=\displaystyle\sum_{k \mathrm{\hspace{2pt} odd},k>0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k) \\
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&= \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k (z-z_0)^{2k+1}}{(2k+1)!},
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\end{array}$$
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as was to be shown. █  
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==References==
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[[Category:Theorem]]
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[[Category:Proven]]

Latest revision as of 03:19, 1 July 2017

Theorem

Let $z_0 \in \mathbb{C}$. The following Taylor series holds: $$\sin(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k z^{2k+1}}{(2k+1)!},$$ where $\sin$ denotes the sine function.

Proof

Using the Taylor series of the exponential function and the definition of $\sin$, $$\begin{array}{ll} \sin(z) &= \dfrac{e^{iz}-e^{-iz}}{2i} \\ &= \dfrac{1}{2i} \left[ \displaystyle\sum_{k=0}^{\infty} \dfrac{i^k (z-z_0)^k}{k!} - \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k i^k (z-z_0)^k}{k!} \right] \\ &= \dfrac{1}{2i} \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k). \end{array}$$ Note that if $k=2n$ is a positive even integer, then $$i^k(1-(-1)^k)=i^{2n}(1-(-1)^{2n})=0,$$ and if $k=2n+1$ is a positive odd integer, then $$i^k(1-(-1)^k)=i^{2n+1}(1-(-1)^{2n+1})=2i(-1)^n.$$ Hence we have derived $$\begin{array}{ll} \sin(z)&=\dfrac{1}{2i} \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k) \\ &=\displaystyle\sum_{k \mathrm{\hspace{2pt} odd},k>0}^{\infty} \dfrac{(z-z_0)^k}{k!}i^k (1-(-1)^k) \\ &= \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k (z-z_0)^{2k+1}}{(2k+1)!}, \end{array}$$ as was to be shown. █

References