Direct 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.

Statement

The comparison test states that if the series

$\sum b_{n}$

is an absolutely convergent series, and if $|a_{n}|\leq |b_{n}|$ for all sufficiently large n , then the series

$\sum a_{n}$

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

$\sum |b_{n}|$

is divergent, and if $|a_{n}|\geq |b_{n}|$ for all sufficiently large n , then the series

$\sum a_{n}$

also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).

Proof

Let $|a_{n}|\leq |b_{n}|$. Let the partial sums of these series be $S_{n}$ and $T_{n}$ respectively i.e.

$S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|,\quad T_{n}=|b_{1}|+|b_{2}|+\ldots +|b_{n}|.$

$T_{n}$ converges as $n\rightarrow \infty$. Denote its limit as $T$. We then have

$S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|\leq |b_{1}|+|b_{2}|+\ldots +|b_{n}|\leq T$

which gives us

$0\leq S_{n}\leq T$

This shows that $S_{n}$ 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

• Convergence tests
• Limit comparison test
• Radius of convergence