Morrie's law

2013-08-03T23:04:57Z

201.141.177.206:

{{Calculus|Series}}

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 <math> \Sigma_n a_n </math> and <math>\Sigma_n b_n</math> with <math> a_n, b_n \geq 0 </math> for all <math> n</math>.

Then if <math> \lim_{n \to \infty} \frac{a_n}{b_n} = c</math> with <math> 0 < c < \infty </math> then either both series converge or both series diverge.

== Proof ==

Because <math> \lim \frac{a_n}{b_n} = c </math> we know that for all <math> \varepsilon </math> there is an integer <math>n_0</math> such that for all <math>n \geq n_0 </math> we have that <math> \left| \frac{a_n}{b_n} - c \right| < \varepsilon </math>, or what is the same

: <math> - \varepsilon < \frac{a_n}{b_n} - c < \varepsilon </math>

: <math> c - \varepsilon < \frac{a_n}{b_n} < c + \varepsilon </math>

: <math> (c - \varepsilon)b_n < a_n < (c + \varepsilon)b_n </math>
As <math> c > 0 </math> we can choose <math> \varepsilon </math> to be sufficiently small such that <math> c-\varepsilon </math> is positive.
So <math> b_n < \frac{1}{c-\varepsilon} a_n </math> and by the [[direct comparison test]], if <math>a_n</math> converges then so does <math> b_n </math>.

Similarly <math> a_n < (c + \varepsilon)b_n </math>, so if <math> b_n </math> converges, again by the direct comparison test, so does <math> a_n </math>.

That is both series converge or both series diverge.

==Example==

We want to determine if the series <math> \sum_{n=1}^{\infty} \frac{1}{n^2 + 2n} </math> converges. For this we compare with the convergent series <math> \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} </math>.

As <math> \lim_{n \to \infty} \frac{1}{n^2 + 2n} \frac{n^2}{1} = 1 > 0 </math> we have that the original series also converges.

==See also==
* [[Convergence tests]]
* [[Direct comparison test]]

== External links ==
* [http://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx Pauls Online Notes on Comparison Test ]

[[Category:Convergence tests]]
[[Category:Articles containing proofs]]

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Morrie's law