Conditional convergence
Property of infinite series
title: "Conditional convergence" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["series-(mathematics)", "integral-calculus", "convergence-(mathematics)", "summability-theory"] description: "Property of infinite series" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Conditional_convergence" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Property of infinite series ::
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_{n=0}^\infty a_n is said to converge conditionally if \lim_{m\rightarrow\infty},\sum_{n=0}^m a_n exists (as a finite real number, i.e. not \infty or -\infty), but \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.
References
- Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).
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