Conditional convergence explained

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 $\backslash sum\_^\backslash infty\; a\_n$ is said to converge conditionally if $\backslash lim\_\backslash ,\backslash sum\_^m\; a\_n$ exists (as a finite real number, i.e. not

inf ty or -inf ty ), but $\backslash sum\_^\backslash infty\; \backslash left|a\_n\backslash right|\; =\; \backslash infty.$

A classic example is the alternating harmonic series given by $$1\; -\; +\; -\; +\; -\; \backslash cdots\; =\backslash sum\backslash limits\_^\backslash infty,$$ 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 Rⁿ can converge.

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

See also

• Absolute convergence
• Unconditional convergence

References

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