Direct comparison test

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 the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known. 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 the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given 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: 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 ∑ a n {displaystyle 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 ∑ c n {displaystyle sum c_{n}} converges absolutely if and only if ∑ | c n | {displaystyle sum |c_{n}|} , a series with nonnegative terms, converges.