Direct comparison test explained

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 \le a _{n} \le 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 \le b _{n} \le 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|} \le |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|} \le |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|} \le |b _{n|} for all positive integers n. Consider the partial sumsS _{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\toinfty} T _{n} = T for some real number T. For all n,0 \le S _{n} = |a _{1|} + |a _{2|} + \ldots + |a _{n|} \le |a _{1|} + \ldots + |a _{n|} + |b _{n+1} | + \ldots = S _{n} + (T - T _{n)} \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,\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 +inf ty or a real number at which f and g each have a vertical asymptote:^[4]

• If the improper integral

\int ^{b} _{a} g(x)dx converges and 0 \le f(x) \le g(x) for a \le x < b , then the improper integral \int ^{b} _{a} f(x)dx also converges with \int ^{b} _{a} f(x)dx \le \int ^{b} _{a} g(x)dx.

• If the improper integral

\int ^{b} _{a} g(x)dx diverges and 0 \le g(x) \le f(x) for a \le x < b , then the improper integral \int ^{b} _{a} 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 (a _{n+1})/(a _{n}) \le (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 (a _{n+1})/(a _{n}) \ge (b _{n+1})/(b _{n}) for all sufficiently large n, then the infinite series \sum a _{n} also diverges.

See also

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

Notes

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.

References

• Book: Ayres. Frank Jr.. Mendelson. Elliott. Elliott Mendelson. Schaum's Outline of Calculus. 4th. McGraw-Hill. New York. 1999. 0-07-041973-6.
• Book: Buck, R. Creighton. Robert Creighton Buck. Advanced Calculus. 2nd. 1965. McGraw-Hill. New York.
• Book: Knopp, Konrad. Konrad Knopp. Infinite Sequences and Series. Dover Publications. New York. 1956. § 3.1. 0-486-60153-6.
• Book: Munem. M. A.. Foulis. D. J.. Calculus with Analytic Geometry. 2nd. 1984. Worth Publishers. 0-87901-236-6.
• Book: Silverman. Herb. Complex Variables. 1975. Houghton Mifflin Company. 0-395-18582-3.
• Book: Whittaker. E. T.. E. T. Whittaker. Watson. G. N.. G. N. Watson. A Course in Modern Analysis. 4th. Cambridge University Press. 1963. § 2.34. 0-521-58807-3.