convergence


convergence
/keuhn verr"jeuhns/, n.
1. an act or instance of converging.
2. a convergent state or quality.
3. the degree or point at which lines, objects, etc., converge.
4. Ophthalm. a coordinated turning of the eyes to bear upon a near point.
5. Physics.
a. the contraction of a vector field.
b. a measure of this.
6. Meteorol. a net flow of air into a given region. Cf. divergence (def. 2).
7. Biol. similarity of form or structure caused by environment rather than heredity.
Also, convergency (for defs. 1-3).
[1705-15; CONVERG(ENT) + -ENCE]

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Mathematical property of infinite series, integrals on unbounded regions, and certain sequences of numbers.

An infinite series is convergent if the sum of its terms is finite. The series 12 + 14 + 18 + 116 + 132 + ... sums to 1 and thus is convergent. The harmonic series 1 + 12 + 13 + 14 + 15 + ... does not converge. An integral calculated over an interval of infinite width, called an improper integral, describes a region that is unbounded in at least one direction. If such an integral converges, the unbounded region it describes has finite area. A sequence of numbers converges to a particular number when the difference between successive terms becomes arbitrarily small. The sequence 0.9, 0.99, 0.999, etc., converges to 1.

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      in mathematics, property (exhibited by certain infinite series and functions (function)) of approaching a limit more and more closely as an argument (variable) of the function increases or decreases or as the number of terms of the series increases.

      For example, the function y = 1/x converges to zero as x increases. Although no finite value of x will cause the value of y to actually become zero, the limiting value of y is zero because y can be made as small as desired by choosing x large enough. The line y = 0 (the x-axis) is called an asymptote of the function.

      Similarly, for any value of x between (but not including) −1 and +1, the series 1 + x + x2 +⋯+ xn converges toward the limit 1/(1 − x) as n, the number of terms, increases. The interval −1 < x < 1 is called the range of convergence of the series; for values of x outside this range, the series is said to diverge.

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Universalium. 2010.

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