Root test

Decision diagram for the root test

The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821).[1] Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series

${\displaystyle \sum _{n=1}^{\infty }a_{n}.}$

the root test uses the number

${\displaystyle C=\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}$

where "lim sup" denotes the limit superior, possibly ∞+. [2] Note that if

${\displaystyle \lim _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}$

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

This test can be used with a power series

${\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-p)^{n}}$

where the coefficients c_n, 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 a_n = c_n(z − p)ⁿ. One then applies the root test to the a_n 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 the Cauchy–Hadamard theorem: the radius of convergence is exactly ${\displaystyle 1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}},}$ taking care that we really mean ∞ if the denominator is 0.

The proof of the convergence of a series Σa_n is an application of the comparison test. If for all n ≥ N (N some fixed natural number) we have ${\displaystyle {\sqrt[{n}]{|a_{n}|}}\leq k<1,}$ then ${\displaystyle |a_{n}|\leq k^{n}<1}$. Since the geometric series ${\displaystyle \sum _{n=N}^{\infty }k^{n}}$ converges so does ${\displaystyle \sum _{n=N}^{\infty }|a_{n}|}$ by the comparison test. Hence Σa_n converges absolutely.

If ${\displaystyle {\sqrt[{n}]{|a_{n}|}}>1}$ for infinitely many n, then a_n fails to converge to 0, hence the series is divergent.

Proof of corollary: For a power series Σa_n = Σc_n(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

${\displaystyle {\sqrt[{n}]{|a_{n}|}}={\sqrt[{n}]{|c_{n}(z-p)^{n}|}}<1,}$

equivalent to

${\displaystyle {\sqrt[{n}]{|c_{n}|}}\cdot |z-p|<1}$

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

${\displaystyle |z-p|<1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}},}$

so ${\displaystyle R\leq 1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}}.}$ Now the only other place where convergence is possible is when

${\displaystyle {\sqrt[{n}]{|a_{n}|}}={\sqrt[{n}]{|c_{n}(z-p)^{n}|}}=1,}$

(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

${\displaystyle R=1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}}.}$

• Ratio test
• Convergent series

• Knopp, Konrad (1956). "§ 3.2". Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0-486-60153-6.
• Whittaker, E. T. & Watson, G. N. (1963). "§ 2.35". A Course in Modern Analysis (fourth 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.