In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series 1 2 + 1 4 + 1 8 + ⋯ {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{8}}+\cdots } is a geometric series with common ratio 1 2 {\displaystyle {\tfrac {1}{2}}} , which converges to the sum of 1 {\displaystyle 1} . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors. While Greek mathematician Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematicians, for example used by Archimedes to calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculating areas of fractals, and various computer science topics. Though geometric series most commonly involve real or complex numbers, there are also important results and applications for matrix-valued geometric series, function-valued geometric series, p {\displaystyle p} -adic number geometric series, and most generally geometric series of elements of abstract algebraic fields, rings, and semirings.
Definition: finite geometric series A geometric series is a series derived from a special type of sequence called a geometric progression. A geometric progression is a sequence obtained from an initial term, producing the next term by multiplying it by a constant from the previous term, and continuing the process with the same constant. Such a constant is called a common ratio. Let a {\displaystyle a} be the initial term and r {\displaystyle r} be the common ratio of a geometric progression. For a finite number of terms, the geometric sequence consists of the elements up to n {\displaystyle n} -th term, which is written as a , a r , a r 2 , a r 3 , … , a r n {\textstyle a,ar,ar^{2},ar^{3},\ldots ,ar^{n}} . Summing all terms above produces a finite geometric series, expressed as
a + a r + a r 2 + a r 3 + ⋯ + a r n = ∑ k = 0 n a r k . {\displaystyle a+ar+ar^{2}+ar^{3}+\cdots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}
When r > 1 {\displaystyle r>1} , it is often called a growth rate or rate of expansion. When 0 < r < 1 {\displaystyle 0<r<1} it is often called a decay rate or shrink rate, where the idea that it is a "rate" comes from interpreting k {\displaystyle k} as a sort of discrete time variable. When an application area has specialized vocabulary for specific types of growth, expansion, shrinkage, and decay, that vocabulary will also often be used to name r {\displaystyle r} parameters of geometric series. In economics, for instance, rates of increase and decrease of price levels are called inflation rates and deflation rates, while rates of increase in values of investments include rates of return and interest rates. For r ≠ 1 {\displaystyle r\neq 1} , the sum of a finite geometric series S n {\displaystyle S_{n}} starting from 0-th term up to n {\displaystyle n} -th term is formulated as:
S n = a ( 1 − r n + 1 ) 1 − r . {\displaystyle S_{n}={\frac {a(1-r^{n+1})}{1-r}}.}
Infinite geometric series and convergence
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