Root test

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In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series

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

It is particularly useful in connection with power series.

Contents

[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|},

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

The root test states that:

  • If C = 1 the test is inconclusive.

[edit] Application to power series

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(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|}}, 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, then an < kn < 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 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 n ≥ N 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 n ≥ N, 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 \ge 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|}}.

[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, 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.