Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

The series of the form $$\sum_{n=0}^{\infty} a_n x^n$$ is called a power series.


\begin{align} \cos x &= \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\\\ \sin x &= \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\\\ e^x &= \sum_{n=0}^{\infty}\frac{x^n}{n!}\\\ \frac{1}{1-x} &= \sum_{n=0}^{\infty} x^n\quad(|x|<1) \end{align}

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