Sine and cosine series

The sine and cosine can be represented in series form by expanding them with a Taylor series about zero. Doing so yields the sine and cosine series:

cos⁡x=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!−…sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!=x−x33!+x55!+…\begin{align} \cos x&=\sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n}}{(2n)!} =1- \frac{x^{2}}{2!}+ \frac{x^{4}}{4!}-\ldots \\ \sin x&=\sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{(2n+1)!}=x- \frac{x^{3}}{3!}+ \frac{x^{5}}{5!}+\ldots \end{align}

Both series are absolutely convergent for all xx.