Root test

From Wikipedia, the free encyclopedia
  (Redirected from Cauchy's radical test)
Jump to: navigation, search

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity

\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},

where a_n are the terms of the series, and states that the series converges absolutely if this quantity is less than one but diverges if it is greater than one. It is particularly useful in connection with power series.

Test[edit]

The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test or Cauchy's radical test. For a series

\sum_{n=1}^\infty a_n.

the root test uses the number

C = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},

where "lim sup" denotes the limit superior, possibly ∞. Note that if

\lim_{n\rightarrow\infty}\sqrt[n]{|a_n|},

converges then it equals C and may be used in the root test instead.

The root test states that:

  • if C < 1 then the series converges absolutely,
  • if C > 1 then the series diverges,
  • if C = 1 and the limit approaches strictly from above then the series diverges,
  • otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).

There are some series for which C = 1 and the series converges, e.g. \textstyle \sum 1/{n^2}, and there are others for which C = 1 and the series diverges, e.g. \textstyle\sum 1/n.

Application to power series[edit]

This test can be used with a power series

f(z) = \sum_{n=0}^\infty c_n (z-p)^n

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(zp)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|}}, taking care that we really mean ∞ if the denominator is 0.

Proof[edit]

The proof of the convergence of a series Σan is an application of the comparison test. If for all nN (N some fixed natural number) we have \sqrt[n]{a_n} < k < 1, then a_n < k^n < 1. Since the geometric series \sum_{n=N}^\infty k^n converges so does \sum_{n=N}^\infty a_n by the comparison test. Absolute convergence in the case of nonpositive an can be proven in exactly the same way using \sqrt[n]{|a_n|}.

If \sqrt[n]{|a_n|} > 1 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 nN we have

\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} < 1,

equivalent to

\sqrt[n]{|c_n|}\cdot|z - p| < 1

for all nN, which implies that in order for the series to converge we must have |z - p| < 1/\sqrt[n]{|c_n|} for all sufficiently large n. This is equivalent to saying

|z - p| < 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},

so R \le 1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}}. Now the only other place where convergence is possible is when

\sqrt[n]{|a_n|} = \sqrt[n]{|c_n(z - p)^n|} = 1,

(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|}}.

See also[edit]

References[edit]

  • 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 Creative Commons Attribution/Share-Alike License.