In mathematics, the 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

<math>\sum_{n=1}^\infty b_n</math>

is an absolutely convergent series and

<math>|a_n|\le |b_n|</math>

for sufficiently large n , then the series

<math>\sum_{n=1}^\infty a_n</math>

converges absolutely. In this case b is said to “dominate” a. If the series ∑|b_n | is divergent and

<math>|a_n|\ge |b_n|</math>

for sufficiently large n , then the series ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  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

• Radius of convergence

Links

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