## Calculus 8th Edition

Maclaurin series is: $\Sigma_{n=0}^{\infty}(-1)^{n}\frac{2^{n}x^{n}}{n!}$ and $R=\infty$
$f(x)=e^{-2x}=\Sigma_{n=0}^{\infty}(-1)^{n}\frac{2^{n}x^{n}}{n!}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{2^{n+1}.x^{n+1}}{(n+1)!}}{\frac{2^{n}x^{n}}{n!!}}|$ $=\lim\limits_{n \to\infty}|(\frac{2x}{n+1})|$ $=0 \lt 1$ Maclaurin series is: $\Sigma_{n=0}^{\infty}(-1)^{n}\frac{2^{n}x^{n}}{n!}$ and $R=\infty$