Power series

A power series is defined as a series that describes some polynomial:

$$\sum _{a=0}^{\infty }{a}_n(z-z_0{)}^n$$

where $z_0\in \mathbb{C}$ is some constant, $a_n\in \mathbb{C}$ (called the coefficients of the series), and $z$ is some variable. It is exactly equal to $f(z)$ defined at all $z$ for which the series converges. If $f(z)$ is equal to the series, then it is analytic for $|z-z_0|<R$.

Power series see important applications in mathematics and engineering, especially as an approximation method, hence our study of them.

Relevant concepts

Some important types of power series are:

• Taylor series (and Maclaurin series)
• Laurent series

And related concepts:

• Series convergence
  • Ratio test
• Continuous-time Fourier series

Table of power series

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$$ $$\frac{1}{1-x}=\sum _{n=0}^{\infty }{x}^n=1+x+x^2+\cdots$$ $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}\cdots$$ $$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$$ $$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}$$