Conditional convergence

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 $\sum_^\infty a_n$ is said to converge conditionally if
$\lim_\,\sum_^m a_n$ exists (as a finite real number, i.e. not $\infty$ or $-\infty$), but $\sum_^\infty \left, a_n\ = \infty.$

A classic example is the alternating harmonic series given by $1 - + - + - \cdots =\sum\limits_^\infty ,$ 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 R^''n'' can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral)
$\int_^ \sin(x^2) dx,$
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)
Series (mathematics)
Integral calculus
Convergence (mathematics)
Summability theory