# Geometric series

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

is geometric, because each term except the first can be obtained by multiplying the previous term by $\frac{1}{2} \$.

Geometric series are one of the simplest examples of infinite series with finite sums. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.

## Contents

### Common ratio

The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. The following table shows several geometric series with different common ratios: