Redirected from Infinite Series
In mathematics, an infinite series is a sum of infinitely many terms. Such a sum can have a finite value; if it has, it is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.
The simplest convergent infinite series is perhaps
This series is a geometric series and mathematicians usually write it as:
Formally, if an infinite series

1) If the series ∑ a_{n} converges, then the sequence (a_{n}) converges to 0 for n→∞; the converse is in general not true.
2) If all the numbers a_{n} are positive and ∑ b_{n} is a convergent series such that a_{n} ≤ b_{n} for all n, then ∑ a_{n} converges as well. Conversely, if all the b_{n} are positive, a_{n} ≥ b_{n} for all n and ∑ b_{n} diverges, then ∑ a_{n} diverges as well.
3) If the a_{n} are positive and there exists a constant C < 1 such that a_{n+1}/a_{n} ≤ C, then ∑ a_{n} converges.
4) If the a_{n} are positive and there exists a constant C < 1 such that (a_{n})^{1/n} ≤ C, then ∑ a_{n} converges.
5) If f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = a_{n} for all n, then ∑ a_{n} converges if and only if the integral ∫_{1}^{∞} f(x) dx exists.
6) A series of the form ∑ (1)^{n} a_{n} (with a_{n} ≥ 0) is called alternating. Such a series converges if the sequence a_{n} is monotone decreasing and converges towards 0. The converse is in general not true.
The series
The geometric series
The sum
If a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Even more: if the a_{n} are real and S is any real number, one can find a reordering so that the reordered series converges with limit S (Riemann).
Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series.
Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with nonconvergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.
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