Problem Set - Power series

## Problem Set - Power series

 Write out the first four terms of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n+1}$. Write out the first four terms of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$. Write out the first four terms of the series $\sum _{n=1}^{\infty }\frac{{\left(x-1\right)}^{n}}{n}$. Write out the first four terms of the series $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{n+2}$. Find the Taylor expansion of ${e}^{x}$ at $x=0$. Find the Taylor expansion of $\mathrm{sinh}x$ at $x=0$. Find the Taylor expansion of $\frac{1}{1-x}$ at $x=0$. Find the Taylor expansion of ${e}^{x}$ at $x=2$. Find the Taylor expansion of $\mathrm{log}x$ at $x=1$. Find the Taylor expansion of $\frac{1}{{x}^{2}}$ at $x=1$. Prove the identity ${e}^{ix}=\mathrm{cos}x+i\mathrm{sin}x$. Prove the identity ${e}^{x}=\mathrm{cosh}x+\mathrm{sinh}x$. Find the radius of convergence of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n+1}$. Find the radius of convergence of the series $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{\left(x+1\right)}^{n}}{{\left(n+1\right)}^{2}}$. Find the radius of convergence of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$. Find the radius of convergence of the series $\sum _{n=1}^{\infty }\frac{{\left(2x-1\right)}^{n}}{\sqrt[3]{n}}$. Find the interval of convergence of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n+1}$. Find the interval of convergence of the series $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{\left(x+1\right)}^{n}}{{\left(n+1\right)}^{2}}$. Find the interval of convergence of the series $\sum _{n=0}^{\infty }\frac{{x}^{n}}{n\text{!}}$. Find the interval of convergence of the series $\sum _{n=1}^{\infty }\frac{{\left(2x-1\right)}^{n}}{\sqrt[3]{n}}$. Find the sum of the series $\sum _{n=1}^{\infty }n{x}^{n-1}$. Find the sum of the series $\sum _{n=0}^{\infty }\frac{{x}^{n+1}}{n+1}$. Find the sum of the series $\sum _{n=1}^{\infty }\frac{{x}^{n}}{n}$. Find the sum of the series $\sum _{n=1}^{\infty }\frac{n}{{3}^{n-1}}$. Find the sum of the series $\sum _{n=1}^{\infty }\frac{1}{n{2}^{n+1}}$. Find the sum of the series $\sum _{n=1}^{\infty }n\left(n-1\right){\left(\frac{1}{4}\right)}^{n}$. Find the power series representation of $\frac{1}{1+2x}$ and determine its radius of convergence. Find the power series representation of $\frac{1}{1+{x}^{2}}$ and determine its radius of convergence. Find the power series representation of $\frac{x}{1+x}$ and determine its radius of convergence. Find the power series representation of $\frac{1}{{\left(1+x\right)}^{2}}$ and determine its radius of convergence. Find the power series representation of $\mathrm{arctan}x$ and determine its radius of convergence. Find the power series representation of $\mathrm{log}\left(2+x\right)$ and determine its radius of convergence. Find the power series representation of $\int {e}^{{x}^{3}}dx$. Find the power series representation of $\int \frac{\mathrm{sinh}x}{x}dx$. Find an infinite series representation of ${\int }_{-1}^{1}\frac{\mathrm{sinh}x}{x}dx$. Find an infinite series representation of ${\int }_{0}^{1}{e}^{{x}^{3}}dx$.