Conditional convergence

Definition

More precisely, a series ∑ n = 0 ∞ a n is said to converge conditionally if lim m → ∞ ∑ n = 0 m a n exists and is a finite number (not ∞ or −∞), but ∑ n = 0 ∞ | a n | = ∞ .

A classic example is the alternating series given by

which converges to 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 sum at all, including ∞ or −∞; see Riemann series theorem.

A typical conditionally convergent integral is that on the non-negative real axis of sin ⁡ ( x 2 ) (see Fresnel integral).