Chapter 11 - Infinite Sequences and Series - Review - Concept Check - Page 824: 11

(a) $\Sigma_{n=0}^\infty x^n=1+x+x^2+...$ .; radius of convergence, $\bf{R}$ is $1$. (b) $e^x=\Sigma_{n=0}^\infty \frac{x^n}{n!}=1+\frac{x}{1!}+\frac{x^2}{2!}+...$ .; radius of convergence, $\bf{R}$ is $\infty$. (c) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}...$ .; radius of convergence, $\bf{R}$ is $\infty$. (d) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}...$ .; radius of convergence, $\bf{R}$ is $\infty$. (e) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)}=x-\frac{x^3}{3}+\frac{x^5}{5}...$ .; radius of convergence, $\bf{R}$ is $1$. (f) $\Sigma_{n=0}^\infty (-1)^{n-1}\frac{x^{n}}{n}=x-\frac{x^2}{2}+\frac{x^3}{3!}...$ .; radius of convergence, $\bf{R}$ is $1$.

(a) $\Sigma_{n=0}^\infty x^n=1+x+x^2+...$ .; radius of convergence, $\bf{R}$ is $1$. (b) $e^x=\Sigma_{n=0}^\infty \frac{x^n}{n!}=1+\frac{x}{1!}+\frac{x^2}{2!}+...$ .; radius of convergence, $\bf{R}$ is $\infty$. (c) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}...$ .; radius of convergence, $\bf{R}$ is $\infty$. (d) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}...$ .; radius of convergence, $\bf{R}$ is $\infty$. (e) $\Sigma_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)}=x-\frac{x^3}{3}+\frac{x^5}{5}...$ .; radius of convergence, $\bf{R}$ is $1$. (f) $\Sigma_{n=0}^\infty (-1)^{n-1}\frac{x^{n}}{n}=x-\frac{x^2}{2}+\frac{x^3}{3!}...$ .; radius of convergence, $\bf{R}$ is $1$.