Cosine series and sine series/R/Definition

For ${\displaystyle {}x\in \mathbb {R} }$, the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}$

is called the cosine series, and the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}$

is called the sine series in ${\displaystyle {}x}$.