## Calculus 8th Edition

Maclaurin's series is: $\Sigma_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n)!}$ and $R=\infty$
$f(x)=xcosx=\Sigma_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{2n!}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{x^{2n+3}}{(2n+2)!}}{\frac{x^{2n+1}}{(2n)!}}|$ $=\lim\limits_{n \to\infty}|\frac{x^{2}}{(2n+2)(2n+1)}|$ $=\lim\limits_{n \to\infty}|\frac{x^{2}}{4n^{2}+6n+2}|$ $=0\lt 1$ Therefore, the Maclaurin's series converges for all values of $x$. Maclaurin's series is: $\Sigma_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n)!}$ and $R=\infty$