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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 + + + + + ... sums to 1 and thus is convergent. The harmonic series 1 + + + + + ... 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.
This entry comes from Encyclopædia Britannica Concise. For the full entry on convergence, visit Britannica.com.