In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
series or
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
is said to be conditionally convergent if it converges, but it does not
converge absolutely.
Definition
More precisely, a series of real numbers
is said to converge conditionally if
exists (as a finite real number, i.e. not
or
), but
A classic example is the
alternating harmonic series given by
which converges to
, but is not absolutely convergent (see
Harmonic series).
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
proved that a conditionally convergent series may be
rearranged to converge to any value at all, including ∞ or −∞; see ''
Riemann series theorem''. The
Lévy–Steinitz theorem identifies the set of values to which a series of terms in R
''n'' can converge.
A typical conditionally convergent integral is that on the non-negative real axis of
(see
Fresnel integral).
See also
*
Absolute convergence
*
Unconditional convergence
References
* Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).
{{series (mathematics)
Mathematical series
Integral calculus
Convergence (mathematics)
Summability theory