Definitions

Comparison test

Comparison test

In mathematics, the comparison test, sometimes called the direct comparison test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. It determines convergence by comparing the terms of the series in question with those of a series whose convergence properties are known.

The comparison test states that if the series

sum_{n=1}^infty b_n

is an absolutely convergent series and

|a_n|le |b_n|

for sufficiently large n , then the series

sum_{n=1}^infty a_n

converges absolutely. In this case b is said to "dominate" a.

If the series

sum_{n=1}^infty |b_n|

is divergent and

|a_n|ge |b_n|

for sufficiently large n , then the series

sum_{n=1}^infty a_n
also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the an  alternate in sign).

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

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