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.

For series

In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms: Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms. Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms: Note that in this last statement, the series \sum a_n 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 \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 a proof of the third 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 for all positive integers n. Consider the partial sums Since \sum b_n converges absolutely, for some real number T. For all n, S_n is a nondecreasing sequence and is nonincreasing. Given m,n > N then both S_n, S_m belong to the interval, whose length T - T_N decreases to zero as N goes to infinity. This shows that is a Cauchy sequence, and so must converge to a limit. Therefore, \sum a_n is absolutely convergent.

For integrals

The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on [a,b) with b either +\infty or a real number at which f and g each have a vertical asymptote:

Ratio comparison test

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: