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.
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- 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.
- 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 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.
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.
The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on with b either or a real number at which f and g each have a vertical asymptote:
- 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 real-valued series, similar to both the direct comparison test above and the ratio test, is called the ratio comparison test:
- If the infinite series converges and , , and for all sufficiently large n, then the infinite series also converges.
- If the infinite series diverges and , , and for all sufficiently large n, then the infinite series also diverges.
- Ayres & Mendelson (1999), p. 401.
- Munem & Foulis (1984), p. 662.
- Silverman (1975), p. 119.
- Buck (1965), p. 140.
- Buck (1965), p. 161.
- Ayres, Frank Jr.; Mendelson, Elliott (1999). Schaum's Outline of Calculus (4th ed.). New York: McGraw-Hill. ISBN 0-07-041973-6.
- Buck, R. Creighton (1965). Advanced Calculus (2nd ed.). New York: McGraw-Hill.
- Knopp, Konrad (1956). Infinite Sequences and Series. New York: Dover Publications. § 3.1. ISBN 0-486-60153-6.
- Munem, M. A.; Foulis, D. J. (1984). Calculus with Analytic Geometry (2nd ed.). Worth Publishers. ISBN 0-87901-236-6.
- Silverman, Herb (1975). Complex Variables. Houghton Mifflin Company. ISBN 0-395-18582-3.
- Whittaker, E. T.; Watson, G. N. (1963). A Course in Modern Analysis (4th ed.). Cambridge University Press. § 2.34. ISBN 0-521-58807-3.