Direct comparison test

In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from the related limit comparison test, is a criterion for convergence or divergence of infinite 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 $\sum b_{n}$ is absolutely convergent and $|a_{n}|\leq |b_{n}|$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also converges absolutely. In this case, the series with larger terms is said to "dominate" the other.
• If the infinite series $\sum b_{n}$ is not absolutely convergent and $|a_{n}|\geq |b_{n}|$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also fails to converge absolutely (although it could still be conditionally convergent if the a_n are not all nonnegative).

Alternatively, the test may be stated in terms of the convergence and divergence of real-valued series with nonnegative terms:

• If the infinite series $\sum b_{n}$ converges and $0\leq a_{n}\leq b_{n}$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also converges.
• If the infinite series $\sum b_{n}$ diverges and $a_{n}\geq b_{n}\geq 0$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also diverges.

These two forms of the test are equivalent for real-valued series because a series $\sum c_{n}$ converges absolutely if and only if $\sum |c_{n}|$, a series with nonnegative terms, converges.

Proof

The proofs of all the statements given above are similar. Here is the proof of the first statement.

Let $\sum a_{n}$ and $\sum b_{n}$ be infinite series such that $\sum b_{n}$ converges absolutely (thus $\sum |b_{n}|$ converges), and without loss of generality assume that $|a_{n}|\leq |b_{n}|$ for all positive integers n. Consider the partial sums

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

Since $\sum b_{n}$ converges absolutely, $\lim _{n\to \infty }T_{n}=T$ for some real number T. The sequence $T_{n}$ is clearly nondecreasing, so $T_{n}\leq T$ for all n. Thus for all n,

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

which implies

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

for all n, as well. This shows that $S_{n}$ is a bounded monotonic sequence and so must converge to a limit. Therefore $\sum a_{n}$ is absolutely convergent.

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
• Integral test for convergence
• Limit comparison test