## Calculus 8th Edition

series has radius of convergence is $4$
$ln(4-x)=ln4-\Sigma_{n=1}^\infty(-1)^{n}\frac{x^{n}}{n4^{n}}$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{\frac{x^{n+1}}{(n+1)4^{n+1}}}{\frac{x^{n}}{n4^{n}}}|$ $\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\frac{x}{(4+4/n)}|$ $=|\frac{x}{4}|$ This converges when $|\frac{x}{4}|\lt 1$ or $=|x| \lt 4$ Thus, the series has radius of convergence is $4$ .