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 given by

which converges to , but is not absolutely convergent (see Harmonic series).

The simplest examples of conditionally convergent series (including the one above) are the alternating 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 .

See also


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