{
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  "slug": "conditional-convergence",
  "title": "Conditional convergence",
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  "lead": "In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.",
  "body": "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).",
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    "title": "Conditional convergence",
    "author": [
      {
        "family": "Anticloud FZ LLE",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/conditional-convergence",
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          14
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