Direct comparison test
Part of a series of articles about  
Calculus  





Specialized


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.
Contents
For series
In calculus, the comparison test for series typically consists of a pair of statements about infinite series with nonnegative (realvalued) terms:^{[1]}
 If the infinite series converges and for all sufficiently large n (that is, for all for some fixed value N), then the infinite series also converges.
 If the infinite series diverges and for all sufficiently large n, then the infinite series also diverges.
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]}
 If the infinite series is absolutely convergent and for all sufficiently large n, then the infinite series is also absolutely convergent.
 If the infinite series is not absolutely convergent and for all sufficiently large n, then the infinite series is also not absolutely convergent.
Note that in this last statement, the series could still be conditionally convergent; for realvalued series, this could happen if the a_{n} are not all nonnegative.
The second pair of statements are equivalent to the first in the case of realvalued series because converges absolutely if and only if , 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 and 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, for some real number T. For all n,
is a nondecreasing sequence and is nonincreasing. Given then both belong to the interval , whose length decreases to zero as goes to infinity. This shows that is a Cauchy sequence, and so must converge to a limit. Therefore, is absolutely convergent.
For integrals
The comparison test for integrals may be stated as follows, assuming continuous realvalued functions f and g on with b either or a real number at which f and g each have a vertical asymptote:^{[4]}
 If the improper integral converges and for , then the improper integral also converges with
 If the improper integral diverges and for , then the improper integral also diverges.
Ratio comparison test
Another test for convergence of realvalued series, similar to both the direct comparison test above and the ratio test, is called the ratio comparison test:^{[5]}
 If the infinite series converges and , , and for all sufficiently large n, then the infinite series also converges.
See also
 Convergence tests
 Convergence (mathematics)
 Dominated convergence theorem
 Integral test for convergence
 Limit comparison test
 Monotone convergence theorem
Notes
References
 Ayres, Frank Jr.; Mendelson, Elliott (1999). Schaum's Outline of Calculus (4th ed.). New York: McGrawHill. ISBN 0070419736.
 Buck, R. Creighton (1965). Advanced Calculus (2nd ed.). New York: McGrawHill.
 Knopp, Konrad (1956). Infinite Sequences and Series. New York: Dover Publications. § 3.1. ISBN 0486601536.
 Munem, M. A.; Foulis, D. J. (1984). Calculus with Analytic Geometry (2nd ed.). Worth Publishers. ISBN 0879012366.
 Silverman, Herb (1975). Complex Variables. Houghton Mifflin Company. ISBN 0395185823.
 Whittaker, E. T.; Watson, G. N. (1963). A Course in Modern Analysis (4th ed.). Cambridge University Press. § 2.34. ISBN 0521588073.