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 Rn can converge.
A typical conditionally convergent integral is that on the non-negative real axis of (see Fresnel integral).
- Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).