Series

Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.

Definition

In mathematics, given an infinite sequence of numbers {an}, a series is, informally speaking, the result of adding all those terms together: a1 + a2 + a3 + ···. These can be written more compactly using the summation symbol ∑. An example is the famous series of Fibonacci numbers.

Types of Series

There are several types of series, including:

• Arithmetic series: A series in which each term after the first is obtained by adding a constant difference to the preceding term.
• Geometric series: A series with a constant ratio between successive terms.
• Harmonic series: A series in which the nth term is the reciprocal of n.
• Power series: A series of the form ∑anxn, where x is a variable and the coefficients an are constants.

Convergence

The concept of convergence is central to the understanding of series. A series is said to converge if the sequence of its partial sums tends to a limit; that means that the partial sums become closer and closer to a certain number when the number of their terms increases.

Applications

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.

See also

• Sequence
• Summation
• Fibonacci numbers
• Arithmetic series
• Geometric series
• Harmonic series
• Power series
• Convergence (mathematics)
• Combinatorics

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