There are several kinds of series expansions, listed below.

A Taylor series is a power series based on a function's derivatives at a single point.^[3] More specifically, if a function ${\displaystyle f:U\to \mathbb {R} }$  is infinitely differentiable around a point ${\displaystyle x_{0}}$ , then the Taylor series of f around this point is given by

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}$ 

under the convention ${\displaystyle 0^{0}:=1}$ .^[3][4] The Maclaurin series of f is its Taylor series about ${\displaystyle x_{0}=0}$ .^[5][4]

A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form ${\textstyle \sum _{k=-\infty }^{\infty }c_{k}(z-a)^{k}}$  and converges in an annulus.^[6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.

A general Dirichlet series is a series of the form ${\textstyle \sum _{n=1}^{\infty }a_{n}e^{-\lambda _{n}s}.}$  One important special case of this is the ordinary Dirichlet series ${\textstyle \sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}$ ^[7] Used in number theory.^{[citation needed]}

A Fourier series is an expansion of periodic functions as a sum of many sine^{[broken anchor]} and cosine^{[broken anchor]} functions.^[8] More specifically, the Fourier series of a function ${\displaystyle f(x)}$  of period ${\displaystyle 2L}$  is given by the expression$${\displaystyle a_{0}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left({\frac {n\pi x}{L}}\right)+b_{n}\sin \left({\frac {n\pi x}{L}}\right)\right]}$$ where the coefficients are given by the formulae^[8][9]$${\displaystyle {\begin{aligned}a_{n}&:={\frac {1}{L}}\int _{-L}^{L}f(x)\cos \left({\frac {n\pi x}{L}}\right)dx,\\b_{n}&:={\frac {1}{L}}\int _{-L}^{L}f(x)\sin \left({\frac {n\pi x}{L}}\right)dx.\end{aligned}}}$$ 

• In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.^{[citation needed]}

• Newtonian series^{[citation needed]}

• Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.^{[citation needed]}

• Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.^{[citation needed]}

• The Stirling series$${\displaystyle {\text{Ln}}\Gamma \left(z\right)\sim \left(z-{\tfrac {1}{2}}\right)\ln z-z+{\tfrac {1}{2}}\ln \left(2\pi \right)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{2k(2k-1)z^{2k-1}}}}$$ is an approximation of the log-gamma function.^[10]

The following is the Taylor series of ${\displaystyle e^{x}}$ :$${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}...}$$ ^[11][12]

The Dirichlet series of the Riemann zeta function is$${\displaystyle \zeta (s):=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+\cdots }$$ ^[7]