Geometric series

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In mathematics, a geometric series is a series in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric sequence forms a geometric series. The name indicates that each term is the geometric mean of its two neighbouring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighbouring terms.

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. 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 ${\displaystyle 1/2}$.

Geometric series have been studied in mathematics from at least the time of Euclid and Archimedes, who evaluated and proved the sum of a geometric series using geometric methods, and had an important role in the early development of calculus. They serve as an introduction to frequently used mathematical tools such as the Taylor series, the Fourier series, and the matrix exponential.