Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

Definition
More precisely, a series of real numbers $\sum_{n=0}^\infty a_n$ is said to converge conditionally if $\lim_{m\rightarrow\infty}\,\sum_{n=0}^m a_n$  exists (as a finite real number, i.e. not $$\infty$$ or $$-\infty$$), but $\sum_{n=0}^\infty \left|a_n\right| = \infty.$

A classic example is the alternating harmonic series 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).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including &infin; or &minus;&infin;; 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 $\sin (x^2)$ (see Fresnel integral).