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 limit comparison test (LCT) (in contrast with the related
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 ...
) is a method of testing for the convergence 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 ...
.
Statement
Suppose that we have two series
and
with
for all
.
Then if
with
, then either both series converge or both series diverge.
Proof
Because
we know that for every
there is a positive integer
such that for all
we have that
, or equivalently
:
:
:
As
we can choose
to be sufficiently small such that
is positive.
So
and by the
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 ...
, if
converges then so does
.
Similarly
, so if
diverges, again by the direct comparison test, so does
.
That is, both series converge or both series diverge.
Example
We want to determine if the series
converges. For this we compare it with the convergent series
As
we have that the original series also converges.
One-sided version
One can state a one-sided comparison test by using
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 ...
. Let
for all
. Then if
with
and
converges, necessarily
converges.
Example
Let
and
for all natural numbers
. Now
does not exist, so we cannot apply the standard comparison test. However,
and since
converges, the one-sided comparison test implies that
converges.
Converse of the one-sided comparison test
Let
for all
. If
diverges and
converges, then necessarily
, that is,
. The essential content here is that in some sense the numbers
are larger than the numbers
.
Example
Let
be analytic in the unit disc
and have image of finite area. By Parseval's formula the area of the image of
is proportional to
. Moreover,
diverges. Therefore, by the converse of the comparison test, we have
, that is,
.
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 t ...
*
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 ...
References
Further reading
* Rinaldo B. Schinazi: ''From Calculus to Analysis''. Springer, 2011, , pp
50* Michele Longo and Vincenzo Valori: ''The Comparison Test: Not Just for Nonnegative Series''. Mathematics Magazine, Vol. 79, No. 3 (Jun., 2006), pp. 205–210
JSTOR
* J. Marshall Ash: ''The Limit Comparison Test Needs Positivity''. Mathematics Magazine, Vol. 85, No. 5 (December 2012), pp. 374–375
JSTOR
External links
Pauls Online Notes on Comparison Test
{{Calculus topics
Convergence tests
Articles containing proofs