Direct comparison test
In mathematics, the comparison test, sometimes called the direct comparison test or CQT[citation needed] (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.
Statement
The comparison test consists of the following pair of statements:
- If the infinite series is absolutely convergent and for all sufficiently large n, then the infinite series also converges absolutely. In this case, the series with larger terms is said to "dominate" the other.
- If the infinite series is not absolutely convergent and for all sufficiently large n, then the infinite series also fails to converge absolutely (although it could still be conditionally convergent if the an are not all nonnegative).
Alternatively, the test may be stated in terms of the convergence and divergence of series with nonnegative terms:
- If the infinite series converges and for all sufficiently large n, then the infinite series also converges.
- If the infinite series diverges and for all sufficiently large n, then the infinite series also diverges.
These two forms of the test are equivalent because a series converges absolutely if and only if , a series with nonegative terms, converges.
Proof
Let . Let the partial sums of these series be and respectively i.e.
converges as . Denote its limit as . We then have
which gives us
This shows that 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