Geometric series

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series

${\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }$

is geometric, because each successive term can be obtained by multiplying the previous term by ${\displaystyle 1/2}$. In general, a geometric series is written as ${\displaystyle a+ar+ar^{2}+ar^{3}+...}$, where ${\displaystyle a}$ is the coefficient of each term and ${\displaystyle r}$ is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential.

The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms.