- Comparison test
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In mathematics, the comparison test, sometimes called the direct comparison test or CQT (in contrast with the related limit comparison test) is a criterion for convergence or divergence of a series whose terms are real or complex numbers. The test determines convergence by comparing the terms of the series in question with those of a series whose convergence properties are known.
Contents
Statement
The comparison test states that if the series
is an absolutely convergent series and for sufficiently large n , then the series
converges absolutely. In this case b is said to "dominate" a. If the series
is divergent and for sufficiently large n , then the series
also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an alternate in sign).
Proof
Let . Let the partial sums of these series be Sn and Tn respectively i.e.
Tn converges as . Denote its limit as T. We then have
which gives us
This shows that Sn is a bounded monotonic sequence and must converge to a limit.
References
- Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.1) ISBN 0-486-60153-6
- Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 2.34) ISBN 0-521-58807-3
See also
- Convergence tests
- Limit comparison test
- Radius of convergence
Categories:- Mathematical series
- Convergence tests
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