Power Series Expansion for Sine Function

Theorem
The sine function has the power series expansion:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

Proof
From Derivative of Sine Function:
 * $\dfrac {\mathrm d}{\mathrm dx} \sin x = \cos x$

From Derivative of Cosine Function:
 * $\dfrac {\mathrm d}{\mathrm dx} \cos x = -\sin x$

Hence:

and so for all $m \in \N$:

where $k \in \Z$.

This leads to the Maclaurin series expansion:

From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.