Definitions

# 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

$sum_\left\{n=1\right\}^infty b_n$

is an absolutely convergent series and

$|a_n|le |b_n|$

for sufficiently large n , then the series

$sum_\left\{n=1\right\}^infty a_n$

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

If the series

$sum_\left\{n=1\right\}^infty |b_n|$

is divergent and

$|a_n|ge |b_n|$

for sufficiently large n , then the series

$sum_\left\{n=1\right\}^infty a_n$
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