In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series frac{1}{2} ,+, frac{1}{4} ,+, frac{1}{8} ,+, frac{1}{16} ,+, cdots is geometric, because each term except the first can be obtained by multiplying the previous term by frac{1}{2} . As more terms are added the total gets closer and closer to 1: the sum converges to the limit 1, as illustrated in the following picture. It is common to say simply that "the sum" of this series is 1, or sometimes that the "the sum to infinity" of the series is 1. 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.

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