Infinite series
In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence has defined first and last terms, whereas a series continues indefinitely.
Given an infinite sequence (a1, a2, a3, ...), a series is informally the result of adding all those terms together: a1 + a2 + a3 + ···. These can be written more compactly using the summation symbol ∑.
A value may not always be given to such an infinite sum, and, in this case, the series is said to be divergent. On the other hand, if the partial sum of the first terms tends to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series.
An example is the famous series from Zeno's dichotomy and its mathematical representation:
which is convergent and whose sum is 1.
The terms of the series are often produced according to a rule, such as by a formula, or by an algorithm. To emphasize that there are an infinite number of terms, a series is often called an infinite series. The study of infinite series is a major part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.
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