Conditional convergence

More precisely, a series of real numbers $\backslash sum\_\{n=0\}^\backslash infty\; a\_n$ is said to converge conditionally if

$\backslash lim\_\{m\backslash rightarrow\backslash infty\}\backslash ,\backslash sum\_\{n=0\}^m\; a\_n$ exists (as a finite real number, i.e. not $\backslash infty$ or $-\backslash infty$), but $\backslash sum\_\{n=0\}^\backslash infty\; \backslash left|a\_n\backslash right|\; =\; \backslash infty.$

A classic example is the alternating harmonic series given by $$1\; -\; \{1\; \backslash over\; 2\}\; +\; \{1\; \backslash over\; 3\}\; -\; \{1\; \backslash over\; 4\}\; +\; \{1\; \backslash over\; 5\}\; -\; \backslash cdots\; =\backslash sum\backslash limits\_\{n=1\}^\backslash infty\; \{(-1)^\{n+1\}\; \backslash over\; n\},$$ which converges to $\backslash ln\; (2)$, 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)

$$\backslash int\_\{0\}^\{\backslash infty\}\; \backslash sin(x^2)\; dx,$$

where the integrand oscillates between positive and negative values

indefinitely, but enclosing smaller areas each time.

• Absolute convergence
• Unconditional convergence

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

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Category:Series (mathematics)

Category:Integral calculus

Category:Convergence (mathematics)

Category:Summability theory