Conditional convergence

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

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 ∞ 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 R n can converge. Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral) \int_{0}^{\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).

Text derived from https://en.wikipedia.org/w/index.php?oldid=1352446826 under CC BY-SA 4.0; modifications have been made.