In mathematics, the root test is a test for the convergence of an infinite series. It is particularly useful in connection with power series. It was developed first by Cauchy and is sometimes known as Cauchy's radical test. In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...

If

,

where lim sup denotes the superior limit, then the series converges absolutely if C < 1 and diverges if C > 1. If the terms of the series are the coefficients of a power series around c In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ... In mathematics, a series is a sum of a sequence of terms. ...

then the radius of convergence of the power series is 1/C, including the infinite radius of convergence as the reciprocal of 0. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In...

References

• Knopp, Konrad (1956). Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0486601536. (§ 3.2)
• Whittaker, E. T., and Watson, G. N. (1963). A Course in Modern Analysis, fourth edition. Cambridge University Press. ISBN 0521588073. (§ 2.35)