In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms:[1] Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms.
[2] Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms:[3] Note that in this last statement, the series
could still be conditionally convergent; for real-valued series, this could happen if the an are not all nonnegative.
The second pair of statements are equivalent to the first in the case of real-valued series because
converges absolutely if and only if
, a series with nonnegative terms, converges.
The proofs of all the statements given above are similar.
Here is a proof of the third statement.
be infinite series such that
converges absolutely (thus
converges), and without loss of generality assume that
for all positive integers n. Consider the partial sums Since
converges absolutely,
lim
for some real number T. For all n,
is a nondecreasing sequence and
is nonincreasing.
belong to the interval
is a Cauchy sequence, and so must converge to a limit.
is absolutely convergent.
The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on
or a real number at which f and g each have a vertical asymptote:[4] Another test for convergence of real-valued series, similar to both the direct comparison test above and the ratio test, is called the ratio comparison test:[5]