Essential Concepts

• Given two power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}[/latex] and [latex]\displaystyle\sum _{n=0}^{\infty }{d}_{n}{x}^{n}[/latex] that converge to functions f and g on a common interval I, the sum and difference of the two series converge to [latex]f\pm g[/latex], respectively, on I. In addition, for any real number b and integer [latex]m\ge 0[/latex], the series [latex]\displaystyle\sum _{n=0}^{\infty }b{x}^{m}{c}_{n}{x}^{n}[/latex] converges to [latex]b{x}^{m}f\left(x\right)[/latex] and the series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(b{x}^{m}\right)}^{n}[/latex] converges to [latex]f\left(b{x}^{m}\right)[/latex] whenever bx^m is in the interval I.
• Given two power series that converge on an interval [latex]\left(\text{-}R,R\right)[/latex], the Cauchy product of the two power series converges on the interval [latex]\left(\text{-}R,R\right)[/latex].
• Given a power series that converges to a function f on an interval [latex]\left(\text{-}R,R\right)[/latex], the series can be differentiated term-by-term and the resulting series converges to [latex]{f}^{\prime }[/latex] on [latex]\left(\text{-}R,R\right)[/latex]. The series can also be integrated term-by-term and the resulting series converges to [latex]\displaystyle\int f\left(x\right)dx[/latex] on [latex]\left(\text{-}R,R\right)[/latex].