Integral test for convergence

The integral test applied to the harmonic series. Since the area under the curve

y = 1/ x

for

x \in [1, \infty)

is infinite, the total area of the rectangles must be infinite as well.

Part of a series of articles about

Calculus

Fundamental theorem

Differential

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

Integral

Definitions
Lists of integrals
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae

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

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Divergence Gradient Green's Kelvin–Stokes

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix

Specialized

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

Statement of the test

Consider an integer $N$ and a non-negative, continuous function $f$ defined on the unbounded interval $[N, \infty)$ , on which it is monotone decreasing. Then the infinite series

\sum _{{n=N}}^{\infty }f(n)

converges to a real number if and only if the improper integral

\int _{N}^{\infty }f(x)\,dx

is finite. In other words, if the integral diverges, then the series diverges as well.

Remark

If the improper integral is finite, then the proof also gives the lower and upper bounds

$\int _{N}^{\infty }f(x)\,dx\leq \sum _{{n=N}}^{\infty }f(n)\leq f(N)+\int _{N}^{\infty }f(x)\,dx$

(1)

for the infinite series.

Proof

The proof basically uses the comparison test, comparing the term $f (n)$ with the integral of $f$ over the intervals $[n - 1, n)$ and $[n, n + 1)$ , respectively.

Since $f$ is a monotone decreasing function, we know that

f(x)\leq f(n)\quad {\text{for all }}x\in [n,\infty )

and

f(n)\leq f(x)\quad {\text{for all }}x\in [N,n].

Hence, for every integer $n \geq N$ ,

$\int _{n}^{{n+1}}f(x)\,dx\leq \int _{{n}}^{{n+1}}f(n)\,dx=f(n)$

(2)

and, for every integer $n \geq N + 1$ ,

$f(n)=\int _{{n-1}}^{{n}}f(n)\,dx\leq \int _{{n-1}}^{n}f(x)\,dx.$

(3)

By summation over all $n$ from $N$ to some larger integer $M$ , we get from (2)

\int _{N}^{{M+1}}f(x)\,dx=\sum _{{n=N}}^{M}\underbrace {\int _{n}^{{n+1}}f(x)\,dx}_{{\leq \,f(n)}}\leq \sum _{{n=N}}^{M}f(n)

and from (3)

\sum _{{n=N}}^{M}f(n)\leq f(N)+\sum _{{n=N+1}}^{M}\underbrace {\int _{{n-1}}^{n}f(x)\,dx}_{{\geq \,f(n)}}=f(N)+\int _{N}^{M}f(x)\,dx.

Combining these two estimates yields

\int _{N}^{{M+1}}f(x)\,dx\leq \sum _{{n=N}}^{M}f(n)\leq f(N)+\int _{N}^{M}f(x)\,dx.

Letting $M$ tend to infinity, the bounds in (1) and the result follow.

Applications

The harmonic series

\sum _{{n=1}}^{\infty }{\frac 1n}

diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get

\int_1^M\frac1n\,dn=\ln n\Bigr|_1^M=\ln M\to\infty \quad\text{for }M\to\infty.

Contrary, the series

\zeta(1+\varepsilon)=\sum_{x=1}^\infty \frac1{x^{1+\varepsilon}}

(cf. Riemann zeta function) converges for every $ε > 0$ , because by the power rule

\int _{1}^{M}{\frac 1{x^{{1+\varepsilon }}}}\,dx=-{\frac 1{\varepsilon x^{\varepsilon }}}{\biggr |}_{1}^{M}={\frac 1\varepsilon }{\Bigl (}1-{\frac 1{M^{\varepsilon }}}{\Bigr )}\leq {\frac 1\varepsilon }<\infty \quad {\text{for all }}M\geq 1.

From (1) we get the upper estimate

\zeta(1+\varepsilon)=\sum_{x=1}^\infty \frac1{x^{1+\varepsilon}}\le\frac{1+\varepsilon}\varepsilon,

which can be compared with some of the particular values of Riemann zeta function.

Borderline between divergence and convergence

The above examples involving the harmonic series raise the question, whether there are monotone sequences such that $f (n)$ decreases to 0 faster than $1/ n$ but slower than $1/ n 1+ ε$ in the sense that

\lim _{{n\to \infty }}{\frac {f(n)}{1/n}}=0\quad {\text{and}}\quad \lim _{{n\to \infty }}{\frac {f(n)}{1/n^{{1+\varepsilon }}}}=\infty

for every $ε > 0$ , and whether the corresponding series of the $f (n)$ still diverges. Once such a sequence is found, a similar question can be asked with $f (n)$ taking the role of $1/ n$ , and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.

Using the integral test for convergence, one can show (see below) that, for every natural number $k$ , the series

$\sum _{{n=N_{k}}}^{\infty }{\frac 1{n\ln(n)\ln _{2}(n)\cdots \ln _{{k-1}}(n)\ln _{k}(n)}}$

(4)

still diverges (cf. proof that the sum of the reciprocals of the primes diverges for $k = 1$ ) but

$\sum _{{n=N_{k}}}^{\infty }{\frac 1{n\ln(n)\ln _{2}(n)\cdots \ln _{{k-1}}(n)(\ln _{k}(n))^{{1+\varepsilon }}}}$

(5)

converges for every $ε > 0$ . Here $ln k$ denotes the $k$ -fold composition of the natural logarithm defined recursively by

\ln _{k}(x)={\begin{cases}\ln(x)&{\text{for }}k=1,\\\ln(\ln _{{k-1}}(x))&{\text{for }}k\geq 2.\end{cases}}

Furthermore, $N k$ denotes the smallest natural number such that the $k$ -fold composition is well-defined and $ln k (N k) \geq 1$ , i.e.

N_{k}\geq \underbrace {e^{{e^{{\cdot ^{{\cdot ^{{e}}}}}}}}}_{{k\ e'{\text{s}}}}=e\uparrow \uparrow k

using tetration or Knuth's up-arrow notation.

To see the divergence of the series (4) using the integral test, note that by repeated application of the chain rule

{\frac {d}{dx}}\ln _{{k+1}}(x)={\frac {d}{dx}}\ln(\ln _{k}(x))={\frac 1{\ln _{k}(x)}}{\frac {d}{dx}}\ln _{k}(x)=\cdots ={\frac 1{x\ln(x)\cdots \ln _{k}(x)}},

hence

\int _{{N_{k}}}^{\infty }{\frac {dx}{x\ln(x)\cdots \ln _{k}(x)}}=\ln _{{k+1}}(x){\bigr |}_{{N_{k}}}^{\infty }=\infty .

To see the convergence of the series (5), note that by the power rule, the chain rule and the above result

-{\frac {d}{dx}}{\frac 1{\varepsilon (\ln _{k}(x))^{\varepsilon }}}={\frac 1{(\ln _{k}(x))^{{1+\varepsilon }}}}{\frac {d}{dx}}\ln _{k}(x)=\cdots ={\frac {1}{x\ln(x)\cdots \ln _{{k-1}}(x)(\ln _{k}(x))^{{1+\varepsilon }}}},

hence

\int _{{N_{k}}}^{\infty }{\frac {dx}{x\ln(x)\cdots \ln _{{k-1}}(x)(\ln _{k}(x))^{{1+\varepsilon }}}}=-{\frac 1{\varepsilon (\ln _{k}(x))^{\varepsilon }}}{\biggr |}_{{N_{k}}}^{\infty }<\infty

and (1) gives bounds for the infinite series in (5).

References

Knopp, Konrad, "Infinite Sequences and Series", Dover Publications, Inc., New York, 1956. (§ 3.3) ISBN 0-486-60153-6
Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN 0-521-58807-3
Ferreira, Jaime Campos, Ed Calouste Gulbenkian, 1987, ISBN 972-31-0179-3

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.