Integral test for convergence - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the integral test for convergence is a method used to test infinite series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

of non-negative terms for convergence. An early form of the test of convergence was developed in India

Indian mathematics

Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...

by Madhava in the 14th century, and by his followers at the Kerala School

Kerala School

The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar...

. In Europe, it was later developed by Maclaurin

Colin Maclaurin

Colin Maclaurin was a Scottish mathematician who made important contributions to geometry and algebra. The Maclaurin series, a special case of the Taylor series, are named after him....

and Cauchy

Augustin Louis Cauchy

Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

and is sometimes known as the Maclaurin–Cauchy test.

Statement of the test

Consider an integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

N and a non-negative monotone decreasing

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

function f defined on the unbounded interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

[N, ∞). Then the series

\sum_{n=N}^\infty f(n)

converges if and only if the integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

\int_N^\infty f(x)\,dx

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

Proof

The proof basically uses the comparison test

Comparison test

In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...

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

and

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

hence for every n larger than N

\int_n^{n+1} f(x)\,dx
\le\int_{n}^{n+1} f(n)\,dx\int_{n-1}^{n} f(n)\,dx
\le\int_{n-1}^n f(x)\,dx.

Since the lower estimate is also valid for f(N), we get by summation over all n from N to some larger integer M

\int_N^{M+1}f(x)\,dx\le\sum_{n=N}^Mf(n)\le f(N)+\int_N^M f(x)\,dx.

Letting M tend to infinity, the result follows.

Applications

The harmonic series

Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

\sum_{n=1}^\infty \frac1n

diverges because, using the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

, its derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

, and the fundamental theorem of calculus

Fundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

, we get

\int_1^M\frac1x\,dx=\ln x\Bigr|_1^M=\ln M\to\infty
\quad\text{for }M\to\infty.

Contrary, the series

\sum_{n=1}^\infty \frac1{n^{1+\varepsilon}}

(cf. Riemann zeta function) converges for every ε > 0, because

\int_1^M\frac1{x^{1+\varepsilon}}\,dx-\frac1{\varepsilon x^\varepsilon}\biggr|_1^M=
\frac1\varepsilon\Bigl(1-\frac1{M^\varepsilon}\Bigr)
\le\frac1\varepsilon
\quad\text{for all }M\ge1.

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

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

k, the series

\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots \ln_{k-1}(n)\ln_k(n)}

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

\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots\ln_{k-1}(n)(\ln_k(n))^{1+\varepsilon}}

converges for every ε > 0. Here ln_k denotes the k-fold composition

Function composition

In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

of the natural logarithm defined recursively

Recursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

\ln_k(x)=
\begin{cases}
\ln(x)&\text{for }k=1,\\
\ln(\ln_{k-1}(x))&\text{for }k\ge2.
\end{cases}

Furthermore, N_k denotes the smallest natural number such that the k-fold composition is well-defined and ln_k N_k ≥ 1, i.e.

N_k\ge \underbrace{e^{e^{\cdot^{\cdot^{e}}}}}_{k\ e'\text{s}}=e \uparrow\uparrow k

using tetration

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

or Knuth's up-arrow notation

Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated...

. To see the divergence of the first series using the integral test, note that by repeated application of the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

\frac{d}{dx}\ln_{k+1}(x)\ln_{k+1}(x)\bigr|_{N_k}^\infty=\infty.

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

-\frac{d}{dx}\frac1{\varepsilon(\ln_k(x))^\varepsilon}\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}}

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