Direct comparison test

Differential

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds

Integral

Definitions
Lists of integrals Integral transform Leibniz integral rule
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Multivariable

Advanced

Specialized

Miscellaneous

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.

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:

If the infinite series ∑ b n {\displaystyle \sum b_{n}} $\sum b_{n}$ converges and 0 ≤ a n ≤ b n {\displaystyle 0\leq a_{n}\leq b_{n}} $0\leq a_{n}\leq b_{n}$ for all sufficiently large n (that is, for all n > N {\displaystyle n>N} $n>N$ for some fixed value N), then the infinite series ∑ a n {\displaystyle \sum a_{n}} $\sum a_{n}$ also converges.
If the infinite series ∑ b n {\displaystyle \sum b_{n}} $\sum b_{n}$ diverges and 0 ≤ b n ≤ a n {\displaystyle 0\leq b_{n}\leq a_{n}} $0\leq b_{n}\leq a_{n}$ for all sufficiently large n, then the infinite series ∑ a n {\displaystyle \sum a_{n}} $\sum a_{n}$ also diverges.

Note that the series having larger terms is sometimes said to dominate (or eventually dominate) the series with smaller terms.

Alternatively, the test may be stated in terms of absolute convergence, in which case it also applies to series with complex terms:

If the infinite series ∑ b n {\displaystyle \sum b_{n}} $\sum b_{n}$ is absolutely convergent and | a n | ≤ | b n | {\displaystyle |a_{n}|\leq |b_{n}|} $|a_{n}|\leq |b_{n}|$ for all sufficiently large n, then the infinite series ∑ a n {\displaystyle \sum a_{n}} $\sum a_{n}$ is also absolutely convergent.
If the infinite series ∑ b n {\displaystyle \sum b_{n}} $\sum b_{n}$ is not absolutely convergent and | b n | ≤ | a n | {\displaystyle |b_{n}|\leq |a_{n}|} $|b_{n}|\leq |a_{n}|$ for all sufficiently large n, then the infinite series ∑ a n {\displaystyle \sum a_{n}} $\sum a_{n}$ is also not absolutely convergent.

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

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}} $\sum b_{n}$ converges absolutely, lim n → ∞ T n = T {\displaystyle \lim _{n\to \infty }T_{n}=T} $\lim _{n\to \infty }T_{n}=T$ 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.}

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}} $S_{n}$ is a nondecreasing sequence and S n + ( T − T n ) {\displaystyle S_{n}+(T-T_{n})} $S_{n}+(T-T_{n})$ is nonincreasing. Given m , n > N {\displaystyle m,n>N} $m,n>N$ then both S n , S m {\displaystyle S_{n},S_{m}} $S_{n},S_{m}$ belong to the interval {\displaystyle } $\text{[math]}$ , whose length T − T N {\displaystyle T-T_{N}} $T-T_{N}$ decreases to zero as N {\displaystyle N} $N$ goes to infinity. This shows that ( S n ) n = 1 , 2 , … {\displaystyle (S_{n})_{n=1,2,\ldots }} $(S_{n})_{n=1,2,\ldots }$ is a Cauchy sequence, and so must converge to a limit. Therefore, ∑ a n {\displaystyle \sum a_{n}} $\sum a_{n}$ is absolutely convergent.

For integrals

The comparison test for integrals may be stated as follows, assuming continuous real-valued functions f and g on

If the improper integral ∫ a b g ( x ) d x {\displaystyle \int _{a}^{b}g(x)\,dx} $\int _{a}^{b}g(x)\,dx$ converges and 0 ≤ f ( x ) ≤ g ( x ) {\displaystyle 0\leq f(x)\leq g(x)} $0\leq f(x)\leq g(x)$ for a ≤ x < b {\displaystyle a\leq x<b} $a\leq x<b$ , then the improper integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} $\int _{a}^{b}f(x)\,dx$ 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.} $\int _{a}^{b}f(x)\,dx\leq \int _{a}^{b}g(x)\,dx.$
If the improper integral ∫ a b g ( x ) d x {\displaystyle \int _{a}^{b}g(x)\,dx} $\int _{a}^{b}g(x)\,dx$ diverges and 0 ≤ g ( x ) ≤ f ( x ) {\displaystyle 0\leq g(x)\leq f(x)} $0\leq g(x)\leq f(x)$ for a ≤ x < b {\displaystyle a\leq x<b} $a\leq x<b$ , then the improper integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} $\int _{a}^{b}f(x)\,dx$ 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:

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

v t e Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Graph of a function Linear function Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Order of approximation (ε, δ)-definition of limit
Differential calculus	Derivative Second derivative Partial derivative Differential Differential operator Mean value theorem Notation Leibniz's notation Newton's notation Rules of differentiation linearity Power Sum Chain L'Hôpital's Product General Leibniz's rule Quotient Other techniques Implicit differentiation Inverse functions and differentiation Logarithmic derivative Related rates Stationary points First derivative test Second derivative test Extreme value theorem Maximum and minimum Further applications Newton's method Taylor's theorem Differential equation Ordinary differential equation Partial differential equation Stochastic differential equation
Integral calculus	Antiderivative Arc length Riemann integral Basic properties Constant of integration Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Tangent half-angle substitution Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method Integral equation Integro-differential equation
Vector calculus	Derivatives Curl Directional derivative Divergence Gradient Laplacian Basic theorems Line integrals Green's Stokes' Gauss'
Multivariable calculus	Divergence theorem Geometric Hessian matrix Jacobian matrix and determinant Lagrange multiplier Line integral Matrix Multiple integral Partial derivative Surface integral Volume integral Advanced topics Differential forms Exterior derivative Generalized Stokes' theorem Tensor calculus
Sequences and series	Arithmetico-geometric sequence Types of series Alternating Binomial Fourier Geometric Harmonic Infinite Power Maclaurin Taylor Telescoping Tests of convergence Abel's Alternating series Cauchy condensation Direct comparison Dirichlet's Integral Limit comparison Ratio Root Term
Special functions and numbers	Bernoulli numbers e (mathematical constant) Exponential function Natural logarithm Stirling's approximation
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Differentiation rules List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions Secant Secant cubed List of limits Lists of integrals
Miscellaneous topics	Complex calculus Contour integral Differential geometry Manifold Curvature of curves of surfaces Tensor Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7 exceeds π Regiomontanus' angle maximization problem Steinmetz solid