Editing Conditional convergence

{{Short description|A property of infinite series}}
In [[mathematics]], a [[series (mathematics)|series]] or [[integral]] is said to be '''conditionally convergent''' if it converges, but it does not [[Absolute convergence|converge absolutely]].

==Definition==
More precisely, a series of real numbers <math display="inline">\sum_{n=0}^\infty a_n</math> is said to '''converge conditionally''' if 
<math display="inline">\lim_{m\rightarrow\infty}\,\sum_{n=0}^m a_n</math> exists (as a finite real number, i.e. not <math>\infty</math> or <math>-\infty</math>), but <math display="inline">\sum_{n=0}^\infty \left|a_n\right| = \infty.</math>

A classic example is the [[alternating series|alternating]] harmonic series given by <math display="block">1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum\limits_{n=1}^\infty {(-1)^{n+1}  \over n},</math> which converges to  <math>\ln (2)</math>, but is not absolutely convergent (see [[Harmonic series (mathematics)|Harmonic series]]).

[[Bernhard Riemann]] proved that a conditionally convergent series may be [[permutation|rearranged]] to converge to any value at all, including &infin; or &minus;&infin;; 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 '''R'''<sup>''n''</sup> can converge.

Indefinite integrals may also be conditionally convergent.  A typical example of a conditionally convergent integral is (see [[Fresnel integral]])
<math display='block'> \int_{0}^{\infty} \sin(x^2) dx,</math>
where the integrand oscillates between positive and negative values
indefinitely, but enclosing smaller areas each time.


==See also==
*[[Absolute convergence]]
*[[Unconditional convergence]]

==References==
* Walter Rudin, ''Principles of Mathematical Analysis'' (McGraw-Hill: New York, 1964).

{{series (mathematics)}}
[[Category:Series (mathematics)]]
[[Category:Integral calculus]]
[[Category:Convergence (mathematics)]]
[[Category:Summability theory]]