The simplest convergent infinite series is perhaps
∞ ∑ 2-n = 2 n=0
Formally, if an infinite series
∞ ∑ an n=0
is given with real (or complex) numbers an, we say that the series converges towards S or that its value is S if the limit
N lim ∑ an N→∞ n=0exists and is equal to S. If this is not the case, we say the series diverges.
1) If the series ∑ an converges, then the sequence (an) converges to 0 for n→∞; the converse is in general not true.
2) If all the numbers an are positive and ∑ bn is a convergent series such that an ≤ bn for all n, then ∑ an converges as well. Conversely, if all the bn are positive, an ≥ bn for all n and ∑ bn diverges, then ∑ an diverges as well.
3) If the an are positive and there exists a constant C < 1 such that an+1/an ≤ C, then ∑ an converges.
4) If the an are positive and there exists a constant C < 1 such that (an)1/n ≤ C, then ∑ an converges.
5) If f(x) is a positive montone decreasing function defined on the interval [1, ∞) with f(n) = an for all n, then ∑ an converges if and only if the integral ∫1∞ f(x) dx exists.
6) The series ∑ an of real numbers is called alternating if the signs of the an alternate. Such a series converges if the sequences |an| is monotone decreasing and converges towards 0. The converse is in general not true.
The series
∞ 1 ∑ --- n=1 nrconverges if r > 1 and diverges for r < 1, which can be shown with the integral criterium 5) from above. This series gives rise to the Riemann zeta function.
The geometric series
∞ ∑ zn n=0converges if and only if |z| < 1.
The sum
∞ ∑ an n=0is said to converge absolutely if the series of absolute values
∞ ∑ |an| n=0converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum. If a series is not absolutely convergent, then there is always some reordering of the terms so that the reordered series diverges.
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 non-convergent 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.