Conditional convergence

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 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 ${\displaystyle \infty }$ or ${\displaystyle -\infty }$), but ${\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .}$

A classic example is the alternating harmonic series given by $${\displaystyle 1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n},}$$ which converges to ${\displaystyle \ln(2)}$, 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. Agnew's theorem describes rearrangements that preserve convergence for all convergent series.

The Lévy–Steinitz theorem identifies the set of values to which a series of terms in Rⁿ can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral) $${\displaystyle \int _{0}^{\infty }\sin(x^{2})dx,}$$ where the integrand oscillates between positive and negative values indefinitely, but enclosing smaller areas each time.