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Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series

1, 2, 4, 8

is geometric, because each successive term can be obtained by multiplying the previous term by

2

A geometric series is the sum of the terms of a geometric sequence. It is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

General Form

The general form of a geometric series is:

a + a r + a r^{2} + a r^{3} + \dots = \sum_{n = 0}^{\infty} a r^{n}

where $a$ is the first term and $r$ is the common ratio.

Sum Formula

The sum of a finite geometric series with $n$ terms is given by the formula:

S_{n} = a \frac{1 - r^{n}}{1 - r}

where $S_{n}$ is the sum of the first $n$ terms.

Properties

If the absolute value of the common ratio $r$ is less than 1, the series converges to a finite value.
If the absolute value of $r$ is greater than or equal to 1, the series diverges.
The sum of an infinite geometric series can be found using the formula for the sum of an infinite geometric series:

S_{\infty} = \frac{a}{1 - r}

Applications

Geometric series have various applications in mathematics, physics, engineering, and finance. They are used to model growth and decay processes, calculate interest, analyze circuits, and more.