In
mathematics, a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
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
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 rig ...
proved that a conditionally convergent series may be
rearranged to converge to any value at all, including ∞ or −∞; see ''
Riemann series theorem
In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its term ...
''. 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
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sai ...
*
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 ...
References
* Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).
{{series (mathematics)
Mathematical series
Integral calculus
Convergence (mathematics)
Summability theory