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:^[1]

If the infinite series $\sum b_{n}$ converges and $0\leq a_{n}\leq b_{n}$ for all sufficiently large n (that is, for all $n>N$ for some fixed value N), then the infinite series $\sum a_{n}$ also converges.
If the infinite series $\sum b_{n}$ diverges and $0\leq b_{n}\leq a_{n}$ for all sufficiently large n, then the infinite series $\sum a_{n}$ 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 $\sum b_{n}$ is absolutely convergent and $|a_{n}|\leq |b_{n}|$ for all sufficiently large n, then the infinite series $\sum a_{n}$ is also absolutely convergent.
If the infinite series $\sum b_{n}$ is not absolutely convergent and $|b_{n}|\leq |a_{n}|$ for all sufficiently large n, then the infinite series $\sum a_{n}$ is also not absolutely convergent.

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 a_n 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 $|a_{n}|\leq |b_{n}|$ for all positive integers n. Consider the partial sums

S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|,\ T_{n}=|b_{1}|+|b_{2}|+\ldots +|b_{n}|.

Since $\sum b_{n}$ converges absolutely, $\lim _{n\to \infty }T_{n}=T$ for some real number T. For all n,

0\leq S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|\leq |a_{1}|+\ldots +|a_{n}|+|b_{n+1}|+\ldots =S_{n}+(T-T_{n})\leq T.

$S_{n}$ is a nondecreasing sequence and $S_{n}+(T-T_{n})$ is nonincreasing. Given $m,n>N$ then both $S_{n},S_{m}$ belong to the interval $[S_{N},S_{N}+(T-T_{N})]$ , whose length $T-T_{N}$ decreases to zero as $N$ goes to infinity. This shows that $(S_{n})_{n=1,2,\ldots }$ 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:^[4]

If the improper integral $\int _{a}^{b}g(x)\,dx$ converges and $0\leq f(x)\leq g(x)$ for $a\leq x<b$ , then the improper integral $\int _{a}^{b}f(x)\,dx$ also converges with $\int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.$
If the improper integral $\int _{a}^{b}g(x)\,dx$ diverges and $0\leq g(x)\leq f(x)$ for $a\leq x<b$ , then the improper integral $\int _{a}^{b}f(x)\,dx$ 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:^[5]

If the infinite series $\sum b_{n}$ converges and $a_{n}>0$ , $b_{n}>0$ , and ${\frac {a_{n+1}}{a_{n}}}\leq {\frac {b_{n+1}}{b_{n}}}$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also converges.
If the infinite series $\sum b_{n}$ diverges and $a_{n}>0$ , $b_{n}>0$ , and ${\frac {a_{n+1}}{a_{n}}}\geq {\frac {b_{n+1}}{b_{n}}}$ for all sufficiently large n, then the infinite series $\sum a_{n}$ also diverges.

Notes

^ Ayres & Mendelson (1999), p. 401.
^ Munem & Foulis (1984), p. 662.
^ Silverman (1975), p. 119.
^ Buck (1965), p. 140.
^ Buck (1965), p. 161.

References

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.

[1] Ayres & Mendelson (1999), p. 401.

[2] Munem & Foulis (1984), p. 662.

[3] Silverman (1975), p. 119.

[4] Buck (1965), p. 140.

[5] Buck (1965), p. 161.

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