Wikipedia / Direct comparison test
Wikipedia / 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 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:
for all sufficiently large n for some fixed value N also converges.
for all sufficiently large n also diverges.
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:
for all sufficiently large n is also absolutely convergent.
for all sufficiently large n is also not absolutely convergent.
could still be conditionally convergent; for real-valued series, this could happen if the an are not all nonnegative.
, 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.
converges), and without loss of generality for all positive integers n. Consider the partial sums
for some real number T. For all n,
is a Cauchy sequence is absolutely convergent.
For integrals
The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g with b or a real number at which f and g each have a vertical asymptote:
also diverges.
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:
for all sufficiently large n also converges.
for all sufficiently large n also diverges.
See also
Notes
References
fr:Série convergente#Principe général : règles de comparaison