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

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Definition

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

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 ∞ or −∞; see *Riemann series theorem*. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in **R**^{n} can converge.

A typical conditionally convergent integral is that on the non-negative real axis of ${\textstyle \sin(x^{2})}$ (see Fresnel integral).