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 ...
, convergence tests are methods of testing for the
convergence
Convergence may refer to:
Arts and media Literature
*''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 ...
,
conditional convergence In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if
\lim_\,\s ...
,
absolute convergence
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 s ...
,
interval 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 ...
or divergence of an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
.
List of tests
Limit of the summand
If the limit of the summand is undefined or nonzero, that is
, then the series must diverge. In this sense, the partial sums are
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. He w ...
only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...
this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
Ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert a ...
This is also known as d'Alembert's criterion.
: Suppose that there exists
such that
::
: If ''r'' < 1, then the series is absolutely convergent. If ''r'' > 1, then the series diverges. If ''r'' = 1, the ratio test is inconclusive, and the series may converge or diverge.
Root test
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity
:\limsup_\sqrt
where a_n are the terms of the series, and states that the series converges absolutely if ...
This is also known as the ''n''th root test or Cauchy's criterion.
: Let
::
: where
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
; if the limit exists it is the same value).
: If ''r'' < 1, then the series converges absolutely. If ''r'' > 1, then the series diverges. If ''r'' = 1, the root test is inconclusive, and the series may converge or diverge.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.
Integral test
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
The series can be compared to an integral to establish convergence or divergence. Let