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Conditional convergence

Property of infinite series

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated May 4, 2026

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

01Definition

More precisely, a series of real numbers {\textstyle \sum _{n=0}^{\infty }a_{n} is said to converge conditionally if {\textstyle \lim _{m\rightarrow \infty }\,\sum _{n=0}^{m}a_{n} exists (as a finite real number, i.e. not \infty or -\infty), but {\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .

A classic example is the alternating harmonic series given by 1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n}, which converges to \ln(2), but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem. Agnew's theorem describes rearrangements that preserve convergence for all convergent series.

The Lévy-Steinitz theorem identifies the set of values to which a series of terms in Rn can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral) \int _{0}^{\infty }\sin(x^{2})dx, where the integrand oscillates between positive and negative values indefinitely, but enclosing smaller areas each time.

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This article is adapted from the Wikipedia article “Conditional convergence”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0 ∞ a n {\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}} is said to converge absolutely if ∑ n = 0 ∞ | a n | = L {\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L} for some real number L .

Unconditional convergence

In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent if it converges but different orderings do not all converge to that same value.