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.


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).

See also[edit]


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