Root test
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In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series
It is particularly useful in connection with power series.
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[edit] The test
The root test was developed first by Cauchy and so is sometimes known as the Cauchy root test or Cauchy's radical test. The root test uses the number
![C = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},](http://upload.wikimedia.org/math/d/d/6/dd63ec5bdc7c477cbd771f2c666f5f6f.png)
where "lim sup" denotes the limit superior, possibly ∞.
The root test states that:
- if C < 1 the series converges absolutely, and
- if C > 1 the series diverges.
(If C = 1 the test is inconclusive. There are some series for which C = 1 and the series converges, e.g. 
, and there are others for which C = 1 and the series diverges, e.g. 
).
[edit] Application to power series
This test can be used with a power series
where the coefficients cn, and the center p are complex numbers and the argument z is a complex variable.
The terms of this series would then be given by an = cn(z − p)n. One then applies the root test to the an as above. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately). A corollary of the root test applied to such a power series is that the radius of convergence is exactly ![1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},](http://upload.wikimedia.org/math/4/c/a/4ca2f1b7fe0fe2cccfd716df451b9131.png)
taking care that we really mean ∞ if the denominator is 0.
[edit] Proof
The proof of the convergence of a series Σan is an application of the comparison test. If for all n ≥ N (N some fixed natural number) we have ![\sqrt[n]{a_n} < k < 1,](http://upload.wikimedia.org/math/7/e/a/7ea9c09b91149631f364de1b9cbd0d35.png)
then an < kn < 1. Since the geometric series 
converges so does 
by the comparison test. Absolute convergence in case of nonpositive an can be proven in exactly the same way using ![\sqrt[n]{|a_n|}.](http://upload.wikimedia.org/math/c/3/0/c30daacea8ac1b3a31a2194fb7d68ce8.png)
If ![\sqrt[n]{|a_n|} > 1](http://upload.wikimedia.org/math/1/b/4/1b498ffe6ffcb7f37c73727fc34947fe.png)
for infinitely many n, then an fails to converge to 0, hence the series is divergent.
Proof of corollary: For a power series Σan = Σcn(z − p)n, we see by the above that the series converges if there exists an N such that for all n ≥ N we have
![\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} < 1,](http://upload.wikimedia.org/math/c/2/5/c257cb3b042552c7f7bc9a96b469f37c.png)
equivalent to
![\sqrt[n]{|c_n|}\cdot|z - p| < 1](http://upload.wikimedia.org/math/5/b/9/5b9f7b4ff5ceeeeb10268e186449d8bd.png)
for all n ≥ N, which implies that in order for the series to converge we must have ![|z - p| < 1/\sqrt[n]{|c_n|}](http://upload.wikimedia.org/math/e/6/8/e68ce874e05a49220b5acbd682cfe38a.png)
for all sufficiently large n. This is equivalent to saying
![|z - p| < 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},](http://upload.wikimedia.org/math/8/1/8/818b24e6bf4db8d2b75b6dc307c2cd08.png)
so ![R \ge 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}}.](http://upload.wikimedia.org/math/b/d/0/bd00f8b27195c4bcd6415a000562ecd1.png)
Now the only other place where convergence is possible is when
![\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} = 1,](http://upload.wikimedia.org/math/1/a/2/1a226389f6950ba5e82d45a7a4c928d5.png)
(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so
![R = 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}}.](http://upload.wikimedia.org/math/1/2/e/12e6e3b1f2b881fbb0566f77c84b6501.png)
[edit] See also
[edit] References
- Knopp, Konrad (1956). "§ 3.2". Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0-486-60153-6.
- Whittaker, E. T., and Watson, G. N. (1963). "§ 2.35". A Course in Modern Analysis (fourth edition ed.). Cambridge University Press. ISBN 0-521-58807-3.
This article incorporates material from Proof of Cauchy's root test on PlanetMath, which is licensed under the GFDL.
