Method of testing for the convergence of an infinite series
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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.
Statement
Suppose that we have two series
and
with
for all
. Then if
with
, then either both series converge or both series diverge.[1]
Proof
Because
we know that for every
there is a positive integer
such that for all
we have that
, or equivalently
![{\displaystyle -\varepsilon <{\frac {a_{n}}{b_{n}}}-c<\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7050fd0a371900dea16f48250040fe7ce40fd9)
![{\displaystyle c-\varepsilon <{\frac {a_{n}}{b_{n}}}<c+\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99d76b08cd07e0daeee18f28a80534e166eebe78)
![{\displaystyle (c-\varepsilon )b_{n}<a_{n}<(c+\varepsilon )b_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46b81a5bb70d3b6a7a4247fe031d9f3d4e8acd89)
As
we can choose
to be sufficiently small such that
is positive. So
and by the direct comparison test, 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. 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
References
- ^ Swokowski, Earl (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 516, ISBN 0-87150-341-7
Further reading
- Rinaldo B. Schinazi: From Calculus to Analysis. Springer, 2011, ISBN 9780817682897, 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