In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the ratio test is a
test
Test(s), testing, or TEST may refer to:
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* ''Test'' (2013 film), an American film
* ''Test'' (2014 film), ...
(or "criterion") for the
convergence
Convergence may refer to:
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*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
of a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
:
where each term is a
real
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* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
and is nonzero when is large. The test was first published by
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
The test
![Decision diagram for the ratio test](https://upload.wikimedia.org/wikipedia/commons/4/4a/Decision_diagram_for_the_ratio_test.svg)
The usual form of the test makes use of the
limit
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The ratio test states that:
* if ''L'' < 1 then the series
converges absolutely
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
;
* if ''L'' > 1 then the series
diverges;
* if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if
limit superior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
and
limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let
:
:
.
Then the ratio test states that:
* if ''R'' < 1, the series converges absolutely;
* if ''r'' > 1, the series diverges;
* if
for all large ''n'' (regardless of the value of ''r''), the series also diverges; this is because
is nonzero and increasing and hence does not approach zero;
* the test is otherwise inconclusive.
If the limit ''L'' in () exists, we must have ''L'' = ''R'' = ''r''. So the original ratio test is a weaker version of the refined one.
Examples
Convergent because ''L'' < 1
Consider the series
:
Applying the ratio test, one computes the limit
:
Since this limit is less than 1, the series converges.
Divergent because ''L'' > 1
Consider the series
:
Putting this into the ratio test:
:
Thus the series diverges.
Inconclusive because ''L'' = 1
Consider the three series
:
:
:
The first series (
1 + 1 + 1 + 1 + ⋯
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit (measurement), unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment ...
) diverges, the second one (the one central to the
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
) converges absolutely and the third one (the
alternating harmonic series
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
\sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots.
The first n terms of the series sum to approximately \ln n + \gamma, wher ...
) converges conditionally. However, the term-by-term magnitude ratios
of the three series are respectively
and
. So, in all three cases, one has that the limit
is equal to 1. This illustrates that when ''L'' = 1, the series may converge or diverge, and hence the original ratio test is inconclusive. In such cases, more refined tests are required to determine convergence or divergence.
Proof
![Ratio test proof](https://upload.wikimedia.org/wikipedia/commons/4/47/Ratio_test_proof.svg)
Below is a proof of the validity of the original ratio test.
Suppose that
. We can then show that the series converges absolutely by showing that its terms will eventually become less than those of a certain convergent
geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
. To do this, consider a real number ''r such that''
. This implies that
for sufficiently large ''n''; say, for all ''n'' greater than ''N''. Hence
for each ''n'' > ''N'' and ''i'' > 0, and so
:
That is, the series converges absolutely.
On the other hand, if ''L'' > 1, then
for sufficiently large ''n'', so that the limit of the summands is non-zero. Hence the series diverges.
Extensions for ''L'' = 1
As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.
[
][
][
]
In all the tests below one assumes that Σ''a''
''n'' is a sum with positive ''a''
''n''. These tests also may be applied to any series with a finite number of negative terms. Any such series may be written as:
:
where ''a
N'' is the highest-indexed negative term. The first expression on the right is a partial sum which will be finite, and so the convergence of the entire series will be determined by the convergence properties of the second expression on the right, which may be re-indexed to form a series of all positive terms beginning at ''n''=1.
Each test defines a test parameter (ρ
n) which specifies the behavior of that parameter needed to establish convergence or divergence. For each test, a weaker form of the test exists which will instead place restrictions upon lim
n->∞ρ
n.
All of the tests have regions in which they fail to describe the convergence properties of Σa
n. In fact, no convergence test can fully describe the convergence properties of the series.
This is because if Σa
n is convergent, a second convergent series Σb
n can be found which converges more slowly: i.e., it has the property that lim
n->∞ (b
n/a
n) = ∞. Furthermore, if Σa
n is divergent, a second divergent series Σb
n can be found which diverges more slowly: i.e., it has the property that lim
n->∞ (b
n/a
n) = 0. Convergence tests essentially use the comparison test on some particular family of a
n, and fail for sequences which converge or diverge more slowly.
De Morgan hierarchy
Augustus De Morgan proposed a hierarchy of ratio-type tests
The ratio test parameters (
) below all generally involve terms of the form
. This term may be multiplied by
to yield
. This term can replace the former term in the definition of the test parameters and the conclusions drawn will remain the same. Accordingly, there will be no distinction drawn between references which use one or the other form of the test parameter.
1. d'Alembert's ratio test
The first test in the De Morgan hierarchy is the ratio test as described above.
2. Raabe's test
This extension is due to
Joseph Ludwig Raabe
Joseph Ludwig Raabe (15 May 1801 in Brody, Galicia – 22 January 1859 in Zürich, Switzerland) was a Swiss mathematician.
Life
As his parents were quite poor, Raabe was forced to earn his living from a very early age by giving private lesson ...
. Define:
:
(and some extra terms, see Ali, Blackburn, Feld, Duris (none), Duris2)
The series will:
* Converge when there exists a ''c>''1 such that
for all ''n>N''.
* Diverge when
for all ''n>N''.
* Otherwise, the test is inconclusive.
For the limit version, the series will:
* Converge if
(this includes the case ''ρ'' = ∞)
* Diverge if
.
* If ''ρ'' = 1, the test is inconclusive.
When the above limit does not exist, it may be possible to use limits superior and inferior.
The series will:
* Converge if
* Diverge if
* Otherwise, the test is inconclusive.
=Proof of Raabe's test
=
Defining
, we need not assume the limit exists; if
, then
diverges, while if
the sum converges.
The proof proceeds essentially by comparison with
. Suppose first that
. Of course
if
then
for large
, so the sum diverges; assume then that
. There exists
such that
for all
, which is to say that
. Thus
, which implies that
for
; since
this shows that
diverges.
The proof of the other half is entirely analogous, with most of the inequalities simply reversed. We need a preliminary inequality to use
in place of the simple