# Conditional convergence

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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 given by

which converges to , but is not absolutely convergent (see Harmonic series).

The simplest examples of conditionally convergent series (including the one above) are the alternating series.

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞; see *Riemann series theorem*.

A typical conditionally convergent integral is that on the non-negative real axis of .

## See also

## References

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