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 \scriptstyle\sum\limits_{n=0}^\infty a_n is said to converge conditionally if \scriptstyle\lim\limits_{m\rightarrow\infty}\,\sum\limits_{n=0}^m\,a_n exists and is a finite number (not ∞ or −∞), but \scriptstyle\sum\limits_{n=0}^\infty \left|a_n\right| = \infty.

A classic example is given by

1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum\limits_{n=1}^\infty {(-1)^{n+1}  \over n}

which converges to \ln (2)\,\!, 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 \sin (x^2).

See also[edit]


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