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Concept
Geometric series
Summary
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
is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and 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 complex Fourier series, and the matrix exponential.
The name geometric succession indicates each term is the geometric mean of its two neighboring terms, similar to how the name Arithmetic succession indicates each term is the arithmetic mean of its two neighboring terms.
The geometric series a + ar + ar2 + ar3 + ... is written in expanded form. Every coefficient in the geometric series is the same. In contrast, the power series written as a0 + a1r + a2r2 + a3r3 + ... in expanded form has coefficients ai that can vary from term to term. In other words, the geometric series is a special case of the power series. The first term of a geometric series in expanded form is the coefficient a of that geometric series.
In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as
and a closed form of the geometric series written as
The derivation of the closed form from the expanded form is shown in this article's section. However even without that derivation, the result can be confirmed with long division: a divided by (1 - r) results in a + ar + ar2 + ar3 + ... , which is the expanded form of the geometric series.
It is often a convenience in notation to set the series equal to the sum s and work with the geometric series
s = a + ar + ar2 + ar3 + ar4 + ... in its normalized form
s / a = 1 + r + r2 + r3 + r4 + ... or in its normalized vector form
s / a = [1 1 1 1 1 ...][1 r r2 r3 r4 .
Official source
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