Wurzeltests

Wurzeltests, commonly called the root test in English, is a criterion in real and complex analysis for the convergence of infinite series. It is particularly useful when the terms a_n grow or decay in a way that is conveniently captured by nth roots. The test is also referred to as the Cauchy root test.

Let sum a_n be a series with complex or real terms. Define L = limsup_{n→∞} |a_n|^{1/n}. Then: if L < 1, the series converges absolutely; if L > 1 (or L = ∞), the series diverges; if L = 1, the test is inconclusive. When the limit exists instead of the limsup, the same conclusions hold with that limit in place of L.

In the special case of a power series, sum c_n z^n, the root test implies that the radius of convergence R is given by R = 1 / limsup_{n→∞} |c_n|^{1/n}. The series converges for |z| < R and diverges for |z| > R.

Examples illustrate the criteria: for a_n = (1/2)^n, |a_n|^{1/n} = 1/2, so L = 1/2 < 1 and the series converges absolutely. For a_n = n! z^n, |a_n|^{1/n} = (n!)^{1/n} |z| ~ (n/e)|z|, which tends to ∞ for any z ≠ 0, so the series converges only at z = 0 (radius of convergence 0).

Limitations: the test is inconclusive when L = 1, and other convergence tests may be required. Related criteria include the ratio test and the various convergence tests for power series.