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
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, mat ...
or an
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 ...
. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
For series
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, the comparison test for series typically consists of a pair of statements about infinite series with non-negative (
real-valued
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
) terms:
* If the infinite series
converges and
for all sufficiently large ''n'' (that is, for all
for some fixed value ''N''), then the infinite series
also converges.
* If the infinite series
diverges and
for all sufficiently large ''n'', then the infinite series
also diverges.
Note that the series having larger terms is sometimes said to ''dominate'' (or ''eventually dominate'') the series with smaller terms.
Alternatively, the test may be stated in terms of
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 sai ...
, in which case it also applies to series with
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
terms:
* If the infinite series
is absolutely convergent and
for all sufficiently large ''n'', then the infinite series
is also absolutely convergent.
* If the infinite series
is not absolutely convergent and
for all sufficiently large ''n'', then the infinite series
is also not absolutely convergent.
Note that in this last statement, the series
could still be
conditionally convergent 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_\,\ ...
; for real-valued series, this could happen if the ''a
n'' are not all nonnegative.
The second pair of statements are equivalent to the first in the case of real-valued series because
converges absolutely if and only if
, a series with nonnegative terms, converges.
Proof
The proofs of all the statements given above are similar. Here is a proof of the third statement.
Let
and
be infinite series such that
converges absolutely (thus
converges), and
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indica ...
assume that
for all positive integers ''n''. Consider the
partial sum
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, mat ...
s
:
Since
converges absolutely,
for some real number ''T''. For all ''n'',
:
is a nondecreasing sequence and
is nonincreasing.
Given
then both
belong to the interval