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 $\scriptstyle \sum \limits _{{n=0}}^{\infty }a_{n}$ is said to converge conditionally if $\scriptstyle \lim \limits _{{m\rightarrow \infty }}\,\sum \limits _{{n=0}}^{m}\,a_{n}$ exists and is a finite number (not ∞ or −∞), but $\scriptstyle \sum \limits _{{n=0}}^{\infty }\left|a_{n}\right|=\infty .$

A classic example is the alternating 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 of real numbers 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 Rⁿ can converge.

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

References

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

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Conditional convergence

Definition

See also

References