In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

Definition

is said to converge conditionally

A classic example is the alternating, 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 Rn can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral 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).