In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
More precisely, a series of real numbers
is said to converge conditionally if
exists (as a finite real number, i.e. not
A classic example is the alternating harmonic series given by
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 Rn can converge.
A typical conditionally convergent integral is that on the non-negative real axis of
sin (
(see Fresnel integral).