Английская Википедия:Conditional convergence

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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 <math display="inline">\sum_{n=0}^\infty a_n</math> is said to converge conditionally if <math display="inline">\lim_{m\rightarrow\infty}\,\sum_{n=0}^m a_n</math> exists (as a finite real number, i.e. not <math>\infty</math> or <math>-\infty</math>), but <math display="inline">\sum_{n=0}^\infty \left|a_n\right| = \infty.</math>

A classic example is the alternating harmonic series given by <math display="block">1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots =\sum\limits_{n=1}^\infty {(-1)^{n+1} \over n},</math> which converges to <math>\ln (2)</math>, 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 <math display="inline">\sin (x^2)</math> (see Fresnel integral).

See also

• Absolute convergence
• Unconditional convergence

References

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