Form and Convergence of a Power Series

Learning OutcomesIdentify a power series and provide examples of themDetermine the radius of convergence and interval of convergence of a power seriesForm of a Power SeriesA series of the form[latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\cdots [/latex],where x is a variable and the coefficients cn are constants, is known as a power series. The series[latex]1+x+{x}^{2}+\cdots =\displaystyle\sum _{n=0}^{\infty }{x}^{n}[/latex]is an example of a power series. Since this series is a geometric series with ratio [latex]r=|x|[/latex], we know that it converges if [latex]|x|<1[/latex] and diverges if [latex]|x|\ge 1[/latex].DefinitionA series of the form[latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\cdots [/latex]is a power series centered at [latex]x=0[/latex]. A series of the form[latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}={c}_{0}+{c}_{1}\left(x-a\right)+{c}_{2}{\left(x-a\right)}^{...