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 ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n$ is said to converge conditionally if $\underset{m\to \mathrm{\infty }}{lim}{\sum }_{n=0}^m{a}_n$ exists (as a finite real number, i.e. not $\mathrm{\infty }$ or $-\mathrm{\infty }$), but ${\sum }_{n=0}^{\mathrm{\infty }}\left|a_n\right|=\mathrm{\infty }.$

A classic example is the alternating harmonic series given by $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots =\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n},$$ 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. 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) $${\int }_0^{\mathrm{\infty }}\sin (x^2)dx,$$ where the integrand oscillates between positive and negative values indefinitely, but enclosing smaller areas each time.

See also

• Absolute convergence
• Unconditional convergence

References

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