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

The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
Another geometric series (coefficient a = 4/9 and common ratio r = 1/9) shown as areas of purple squares. The total purple area is S = a / (1 - r) = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the unit square is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.

is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar2 + ar3 + ... , where a is the coefficient of each term and r is the common ratio between adjacent terms. Geometric series are among the simplest examples of infinite series and can serve as an introduction to Taylor series and Fourier series. Geometric series had an important role in the early development of calculus, are used throughout mathematics, and have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

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. The sequence of geometric series terms (without any of the additions) is called a geometric sequence or, equivalently, a geometric progression.

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