Essential Concepts

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