Direct comparison test

In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (real-valued) terms:Ayres & Mendelson (1999), p. 401.

• If the infinite series $\backslash sum\; b\_n$ converges and $0\; \backslash le\; a\_n\; \backslash le\; b\_n$ for all sufficiently large n (that is, for all $n>N$ for some fixed value N), then the infinite series $\backslash sum\; a\_n$ also converges.
• If the infinite series $\backslash sum\; b\_n$ diverges and $0\; \backslash le\; b\_n\; \backslash le\; a\_n$ for all sufficiently large n, then the infinite series $\backslash sum\; a\_n$ also diverges.

Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms.Munem & Foulis (1984), p. 662.

Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms:Silverman (1975), p. 119.

• If the infinite series $\backslash sum\; b\_n$ is absolutely convergent and $|a\_n|\; \backslash le\; |b\_n|$ for all sufficiently large n, then the infinite series $\backslash sum\; a\_n$ is also absolutely convergent.
• If the infinite series $\backslash sum\; b\_n$ is not absolutely convergent and $|b\_n|\; \backslash le\; |a\_n|$ for all sufficiently large n, then the infinite series $\backslash sum\; a\_n$ is also not absolutely convergent.

Note that in this last statement, the series $\backslash sum\; a\_n$ 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 $\backslash sum\; c\_n$ converges absolutely if and only if $\backslash sum\; |c\_n|$, a series with nonnegative terms, converges.

The proofs of all the statements given above are similar. Here is a proof of the third statement.

Let $\backslash sum\; a\_n$ and $\backslash sum\; b\_n$ be infinite series such that $\backslash sum\; b\_n$ converges absolutely (thus $\backslash sum\; |b\_n|$ converges), and without loss of generality assume that $|a\_n|\; \backslash le\; |b\_n|$ for all positive integers n. Consider the partial sums

:$S\_n\; =\; |a\_1|\; +\; |a\_2|\; +\; \backslash ldots\; +\; |a\_n|,\backslash \; T\_n\; =\; |b\_1|\; +\; |b\_2|\; +\; \backslash ldots\; +\; |b\_n|.$

Since $\backslash sum\; b\_n$ converges absolutely, $\backslash lim\_\{n\backslash to\backslash infty\}\; T\_n\; =\; T$ for some real number T. For all n,

:$0\; \backslash le\; S\_n\; =\; |a\_1|\; +\; |a\_2|\; +\; \backslash ldots\; +\; |a\_n|\; \backslash le\; |a\_1|\; +\; \backslash ldots\; +\; |a\_n|\; +\; |b\_\{n+1\}|\; +\; \backslash ldots\; =\; S\_n\; +\; (T-T\_n)\; \backslash le\; T.$

$S\_n$ 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 $(S\_n)\_\{n=1,2,\backslash ldots\}$ is a Cauchy sequence, and so must converge to a limit. Therefore, $\backslash sum\; a\_n$ is absolutely convergent.

The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on $[a,b)$ with b either $+\backslash infty$ or a real number at which f and g each have a vertical asymptote:Buck (1965), p. 140.

• If the improper integral $\backslash int\_a^b\; g(x)\backslash ,dx$ converges and $0\; \backslash le\; f(x)\; \backslash le\; g(x)$ for , then the improper integral $\backslash int\_a^b\; f(x)\backslash ,dx$ also converges with $\backslash int\_a^b\; f(x)\backslash ,dx\; \backslash le\; \backslash int\_a^b\; g(x)\backslash ,dx.$
• If the improper integral $\backslash int\_a^b\; g(x)\backslash ,dx$ diverges and $0\; \backslash le\; g(x)\; \backslash le\; f(x)$ for , then the improper integral $\backslash int\_a^b\; f(x)\backslash ,dx$ also diverges.

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:Buck (1965), p. 161.

• If the infinite series $\backslash sum\; b\_n$ converges and $a\_n>0$, $b\_n>0$, and $\backslash frac\{a\_\{n+1\}\}\{a\_n\}\; \backslash le\; \backslash frac\{b\_\{n+1\}\}\{b\_n\}$ for all sufficiently large n, then the infinite series $\backslash sum\; a\_n$ also converges.
• If the infinite series $\backslash sum\; b\_n$ diverges and $a\_n>0$, $b\_n>0$, and $\backslash frac\{a\_\{n+1\}\}\{a\_n\}\; \backslash ge\; \backslash frac\{b\_\{n+1\}\}\{b\_n\}$ for all sufficiently large n, then the infinite series $\backslash sum\; a\_n$ also diverges.

{{Portal|Mathematics}}

• Convergence tests
• Convergence (mathematics)
• Dominated convergence theorem
• Integral test for convergence
• Limit comparison test
• Monotone convergence theorem

• {{cite book|last1=Ayres|first1=Frank Jr.|last2=Mendelson|first2=Elliott|author-link2=Elliott Mendelson|title=Schaum's Outline of Calculus|edition=4th|publisher=McGraw-Hill|location=New York|date=1999|isbn=0-07-041973-6|url=https://archive.org/details/schaumsoutlineof00ayre_0}}
• {{cite book|last=Buck|first=R. Creighton|author-link=Robert Creighton Buck|title=Advanced Calculus|edition=2nd|date=1965|publisher=McGraw-Hill|location=New York}}
• {{cite book|last=Knopp|first=Konrad|author-link=Konrad Knopp|title=Infinite Sequences and Series|publisher=Dover Publications|location=New York|date=1956|at=§ 3.1|isbn=0-486-60153-6}}
• {{cite book|last1=Munem|first1=M. A.|last2=Foulis|first2=D. J.|title=Calculus with Analytic Geometry|edition=2nd|date=1984|publisher=Worth Publishers|isbn=0-87901-236-6|url=https://archive.org/details/calculuswithanal00mune}}
• {{cite book|last=Silverman|first1=Herb|title=Complex Variables|date=1975|publisher=Houghton Mifflin Company|isbn=0-395-18582-3}}
• {{cite book|last1=Whittaker|first1=E. T.|author-link1=E. T. Whittaker|last2=Watson|first2=G. N.|author-link2=G. N. Watson|title=A Course in Modern Analysis|edition=4th|publisher=Cambridge University Press|date=1963|at=§ 2.34|isbn=0-521-58807-3}}

{{Calculus topics}}

Category:Convergence tests

fr:Série convergente#Principe général : règles de comparaison