In
mathematics, the integral test for convergence is a
method used to test infinite
series of
monotonous
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
* Monotonic ...
terms for
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
* "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that united the four Wei ...
. 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
and a function defined on the unbounded
interval , on which it is
monotone decreasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
. Then the infinite series
:
converges to a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
if and only if the
improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpo ...
:
is finite. In particular, 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
for the infinite series.
Note that if the function
is increasing, then the function
is decreasing and the above theorem applies.
Proof
The proof basically uses the
comparison test, comparing the term with the integral of over the intervals
and , respectively.
The monotonous function
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
almost everywhere. To show this, let
. For every
, there exists by the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of
a
so that
and
:
f(n)\le f(x)\quad\textx\in[N,n">,\infty)
and
:
f(n)\le f(x)\quad\textx\in[N,n
Hence, for every integer ,
and, for every integer ,
By summation over all from to some larger integer , we get from ()
:
\int_N^f(x)\,dx=\sum_^M\underbrace_\le\sum_^Mf(n)
and from ()
:
\sum_^Mf(n)=f(N)+\sum_^Mf(n)\le f(N)+\sum_^M\underbrace_=f(N)+\int_N^M f(x)\,dx.
Combining these two estimates yields
:
\int_N^f(x)\,dx\le\sum_^Mf(n)\le f(N)+\int_N^M f(x)\,dx.
Letting tend to infinity, the bounds in () and the result follow.
Applications
The harmonic series (mathematics)">harmonic series
:
\sum_^\infty \frac 1 n
diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get
:
\int_1^M \frac 1 n\,dn = \ln n\Bigr, _1^M = \ln M \to\infty
\quad\textM\to\infty.
On the other hand, the series
:
\zeta(1+\varepsilon)=\sum_^\infty \frac1
(cf.
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
)
converges for every , because by the
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated u ...
:
\int_1^M\frac1\,dn
= \left. -\frac 1 \_1^M=
\frac 1 \varepsilon \left(1-\frac 1 \right)
\le \frac 1 \varepsilon < \infty
\quad\textM\ge1.
From () we get the upper estimate
:
\zeta(1+\varepsilon)=\sum_^\infty \frac 1 \le \frac\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 decreases to 0 faster than but slower than in the sense that
:
\lim_\frac=0
\quad\text\quad
\lim_\frac=\infty
for every , and whether the corresponding series of the still diverges. Once such a sequence is found, a similar question can be asked with taking the role of , 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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
, the series
still diverges (cf.
proof that the sum of the reciprocals of the primes diverges
The sum of the reciprocals of all prime numbers diverges; that is:
\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty
This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-centur ...
for ) but
converges for every . Here denotes the -fold
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of the natural logarithm defined
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
by
:
\ln_k(x)=
\begin
\ln(x)&\textk=1,\\
\ln(\ln_(x))&\textk\ge2.
\end
Furthermore, denotes the smallest natural number such that the -fold composition is well-defined and , i.e.
:
N_k\ge \underbrace_=e \uparrow\uparrow k
using
tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as r ...
or
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.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperat ...
.
To see the divergence of the series () using the integral test, note that by repeated application of the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
:
\frac\ln_(x)
=\frac\ln(\ln_k(x))
=\frac1\frac\ln_k(x)
=\cdots
=\frac1,
hence
:
\int_^\infty\frac
=\ln_(x)\bigr, _^\infty=\infty.
To see the convergence of the series (), note that by the
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated u ...
, the chain rule and the above result
:
-\frac\frac1
=\frac1\frac\ln_k(x)
=\cdots
=\frac,
hence
:
\int_^\infty\frac
=-\frac1\biggr, _^\infty<\infty
and () gives bounds for the infinite series in ().
See also
*
Convergence tests
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n.
List of tests
Limit of the summand
If ...
*
Convergence (mathematics)
*
Direct comparison test
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 ...
*
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
*
Euler-Maclaurin formula
*
Limit comparison test
*
Monotone convergence theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Inf ...
References
*
Knopp, Konrad, "Infinite Sequences and Series",
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...
, Inc., New York, 1956. (§ 3.3)
*
Whittaker, E. T., and Watson, G. N., ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1963. (§ 4.43)
* Ferreira, Jaime Campos, Ed Calouste Gulbenkian, 1987,
{{Calculus topics
Augustin-Louis Cauchy
Integral calculus
Convergence tests
Articles containing proofs