Comparison test

Comparison test

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

\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).

Proof

Let |a_n| \le |b_n|. Let the partial sums of these series be Sn and Tn respectively i.e.

S_n = |a_1| + |a_2| + \ldots + |a_n|, \quad T_n = |b_1| + |b_2| + \ldots + |b_n|.

Tn converges as n \rightarrow \infty. Denote its limit as T. We then have

 S_n = |a_1| + |a_2| + \ldots + |a_n| \le |b_1| + |b_2| + \ldots + |b_n| \le T

which gives us

 0 \leq S_n \le T

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


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