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Absolute SummableEdit

Absolute Summable is a fundamental concept in mathematical analysis describing when a series behaves particularly well in terms of convergence. It concerns series whose terms, when taken in absolute value, sum to a finite number. This property is stronger than ordinary convergence of the series itself and has important implications for rearrangements, function series, and practical computations.

In plain terms, a series ∑ a_n is absolutely summable if ∑ |a_n| converges. When this happens, not only does the original series ∑ a_n converge, but it does so in a way that is robust under changes to the order of summation. This robustness is a hallmark of absolute summability and is central to many results in real analysis and its applications.

Definition

Let {a_n} be a sequence of real or complex numbers. The series ∑ a_n is said to be absolutely summable if the series of nonnegative terms ∑ |a_n| converges, i.e.,

  • ∑ |a_n| < ∞.

If ∑ |a_n| converges, then the original series ∑ a_n also converges, by the triangle inequality and Cauchy criterion. Conversely, if ∑ a_n converges but ∑ |a_n| diverges, the series is said to converge conditionally rather than absolutely.

For series of functions, the notion extends naturally: a series ∑ f_n(x) is absolutely summable on a domain D if ∑ |f_n(x)| converges for every x in D (and often with a uniform or almost-everywhere sense, depending on the context).

Encyclopedia links: series, convergence, absolute convergence.

Basic properties

  • If ∑ |a_n| converges, then a_n → 0. This is a direct consequence of the convergence of the tail sums.
  • Absolute summability implies ordinary convergence: ∑ a_n converges to some finite value.
  • Rearrangement independence: If ∑ |a_n| converges, then any rearrangement of the terms yields the same sum. This is a manifestation of unconditional convergence for absolutely convergent series.
  • Stability under addition: If ∑ a_n and ∑ b_n are absolutely convergent, then ∑ (a_n + b_n) is absolutely convergent, with sum equal to the sum of the two original sums.
  • In function spaces, absolute summability interacts with uniform behavior: a series of functions ∑ f_n can converge uniformly if ∑ ||f_n||∞ converges (the Weierstrass M-test gives a concrete criterion).

Encyclopedia links: Weierstrass M-test, uniform convergence, series, absolute convergence.

Examples

  • Absolutely summable: ∑ 1/n^p with p > 1. Here ∑ |1/n^p| converges, so the series is absolutely summable and hence convergent.
  • Absolutely summable with alternating signs: ∑ (-1)^n / n^2. Since |(-1)^n / n^2| = 1/n^2 and ∑ 1/n^2 converges, the series is absolutely summable.
  • Not absolutely summable: ∑ 1/n (the harmonic series). The sum of absolute values diverges, so the series is not absolutely summable; it converges conditionally only in the alternating case, such as ∑ (-1)^{n+1}/n.

Encyclopedia links: harmonic series, p-series, alternating series test.

Tests for absolute summability

  • Ratio test (applied to ∑ |a_n|): If limsup_{n→∞} |a_{n+1}/a_n| < 1, then ∑ |a_n| converges.
  • Root test: If limsup_{n→∞} (|a_n|)^{1/n} < 1, then ∑ |a_n| converges.
  • Comparison test: If there exists a convergent nonnegative series ∑ b_n with |a_n| ≤ b_n for all large n, then ∑ |a_n| converges.
  • Integral test (for suitable monotone positive sequences): If f(n) = |a_n| corresponds to a positive decreasing function f, then ∑ |a_n| converges iff the improper integral ∫ f(x) dx converges.
  • These tests are typically stated for ∑ |a_n| and then translated back to the original series ∑ a_n.

Encyclopedia links: ratio test, root test, comparison test, integral test.

Relation to convergence and rearrangements

  • Absolute convergence guarantees convergence regardless of the order in which terms are added. In particular, any rearrangement of an absolutely convergent series converges to the same sum.
  • There is a sharp contrast with conditional convergence: a series can converge without its absolute values summing to a finite number, and in such cases rearrangements can alter the sum or even cause divergence.
  • The Riemann rearrangement theorem highlights this distinction: for conditionally convergent series, rearrangement can produce any real number or divergence, whereas absolutely convergent series are unconditionally convergent.
  • In analysis of series of functions, absolute convergence can lead to stronger forms of convergence, such as uniform convergence on compact sets under appropriate hypotheses.

Encyclopedia links: Riemann rearrangement theorem, unconditional convergence, conditional convergence.

Applications

  • Fourier analysis: when a Fourier coefficient series is absolutely summable, the corresponding series behaves nicely, with strong convergence and favorable error properties for representing the target function.
  • Numerical computation: absolute summability provides stable numerical behavior, since tail errors can be tightly bounded by the sum of absolute values.
  • Signal processing and related disciplines: absolute summability helps in guaranteeing that infinite impulse responses do not accumulate unboundedly, aiding in the design of stable filters.
  • Functional analysis and operator theory: absolute summability conditions appear in the study of series in Banach spaces and in criteria for boundedness and compactness of certain operators.

Encyclopedia links: Fourier series, Banach space.

See also