# 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
is said to **converge conditionally** if
exists and is a finite number (not ∞ or −∞), but

A classic example is the alternating series given by

which converges to , 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*. 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).

## References

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