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

Note that in this last statement, the series ∑ a n {\displaystyle \sum a_{n}} 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 ∑ c n {\displaystyle \sum c_{n}} converges absolutely if and only if ∑ | c n | {\displaystyle \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 ∑ a n {\displaystyle \sum a_{n}} and ∑ b n {\displaystyle \sum b_{n}} be infinite series such that ∑ b n {\displaystyle \sum b_{n}} converges absolutely (thus ∑ | b n | {\displaystyle \sum |b_{n}|} converges), and without loss of generality assume that | a n | ≤ | b n | {\displaystyle |a_{n}|\leq |b_{n}|} for all positive integers n. Consider the partial sums

S n = | a 1 | + | a 2 | + … + | a n | ,   T n = | b 1 | + | b 2 | + … + | b n | . {\displaystyle S_{n}=|a_{1}|+|a_{2}|+\ldots +|a_{n}|,\ T_{n}=|b_{1}|+|b_{2}|+\ldots +|b_{n}|.}

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

0 ≤ S n = | a 1 | + | a 2 | + … + | a n | ≤ | a 1 | + … + | a n | + | b n + 1 | + … = S n + ( T − T n ) ≤ T . {\displaystyle 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 {\displaystyle S_{n}} is a nondecreasing sequence and S n + ( T − T n ) {\displaystyle S_{n}+(T-T_{n})} is nonincreasing. Given m , n > N {\displaystyle m,n>N} then both S n , S m {\displaystyle S_{n},S_{m}} belong to the interval [ S N , S N + ( T − T N ) ] {\displaystyle [S_{N},S_{N}+(T-T_{N})]} , whose length T − T N {\displaystyle T-T_{N}} decreases to zero as N {\displaystyle N} goes to infinity. This shows that ( S n ) n = 1 , 2 , … {\displaystyle (S_{n})_{n=1,2,\ldots }} is a Cauchy sequence, and so must converge to a limit. Therefore, ∑ a n {\displaystyle \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 ) {\displaystyle [a,b)} with b either + ∞ {\displaystyle +\infty } or a real number at which f and g each have a vertical asymptote:⁴

• If the improper integral ∫ a b g ( x ) d x {\displaystyle \int _{a}^{b}g(x)\,dx} converges and 0 ≤ f ( x ) ≤ g ( x ) {\displaystyle 0\leq f(x)\leq g(x)} for a ≤ x < b {\displaystyle a\leq x<b} , then the improper integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} also converges with ∫ a b f ( x ) d x ≤ ∫ a b g ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.}
• If the improper integral ∫ a b g ( x ) d x {\displaystyle \int _{a}^{b}g(x)\,dx} diverges and 0 ≤ g ( x ) ≤ f ( x ) {\displaystyle 0\leq g(x)\leq f(x)} for a ≤ x < b {\displaystyle a\leq x<b} , then the improper integral ∫ a b f ( x ) d x {\displaystyle \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 ∑ b n {\displaystyle \sum b_{n}} converges and a n > 0 {\displaystyle a_{n}>0} , b n > 0 {\displaystyle b_{n}>0} , and a n + 1 a n ≤ b n + 1 b n {\displaystyle {\frac {a_{n+1}}{a_{n}}}\leq {\frac {b_{n+1}}{b_{n}}}} for all sufficiently large n, then the infinite series ∑ a n {\displaystyle \sum a_{n}} also converges.
• If the infinite series ∑ b n {\displaystyle \sum b_{n}} diverges and a n > 0 {\displaystyle a_{n}>0} , b n > 0 {\displaystyle b_{n}>0} , and a n + 1 a n ≥ b n + 1 b n {\displaystyle {\frac {a_{n+1}}{a_{n}}}\geq {\frac {b_{n+1}}{b_{n}}}} for all sufficiently large n, then the infinite series ∑ a n {\displaystyle \sum a_{n}} also diverges.

See also

• Mathematics portal

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

Notes

• 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. {{cite book}}: ISBN / Date incompatibility (help)
• 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. {{cite book}}: ISBN / Date incompatibility (help)

References

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. ↩