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 root test is a criterion for the
convergence (a
convergence test) of an
infinite series. It depends on the quantity
:
where
are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
.
Root test explanation
The root test was developed first by
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
who published it in his textbook
Cours d'analyse
''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...
(1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series
:
the root test uses the number
:
where "lim sup" denotes the
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 ...
, possibly ∞+. Note that if
:
converges then it equals ''C'' and may be used in the root test instead.
The root test states that:
* if ''C'' < 1 then the series
converges absolutely,
* if ''C'' > 1 then the series
diverges,
* if ''C'' = 1 and the limit approaches strictly from above then the series diverges,
* otherwise the test is inconclusive (the series may diverge, converge absolutely or
converge conditionally).
There are some series for which ''C'' = 1 and the series converges, e.g.
, and there are others for which ''C'' = 1 and the series diverges, e.g.
.
Application to power series
This test can be used with a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
:
where the coefficients ''c''
''n'', and the center ''p'' are
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 fo ...
s and the argument ''z'' is a complex variable.
The terms of this series would then be given by ''a''
''n'' = ''c''
''n''(''z'' − ''p'')
''n''. One then applies the root test to the ''a''
''n'' as above. Note that sometimes a series like this is called a power series "around ''p''", because the
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
is the radius ''R'' of the largest interval or disc centred at ''p'' such that the series will converge for all points ''z'' strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately). A
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of the root test applied to such a power series is the
Cauchy–Hadamard theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cau ...
: the radius of convergence is exactly
taking care that we really mean ∞ if the denominator is 0.
Proof
The proof of the convergence of a series Σ''a''
''n'' is an application of the
comparison test. If for all ''n'' ≥ ''N'' (''N'' some fixed
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 ...
) we have
, then
. Since the
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 suc ...
converges so does
by the comparison test. Hence Σ''a''
''n'' converges absolutely.
If
for infinitely many ''n'', then ''a''
''n'' fails to converge to 0, hence the series is divergent.
Proof of corollary:
For a power series Σ''a''
''n'' = Σ''c''
''n''(''z'' − ''p'')
''n'', we see by the above that the series converges if there exists an ''N'' such that for all ''n'' ≥ ''N'' we have
:
equivalent to
:
for all ''n'' ≥ ''N'', which implies that in order for the series to converge we must have