Wurzeltest

Wurzeltest, also known as the root test or Cauchy root test, is a convergence test for infinite series and, by extension, for power series. For a series sum a_n with complex or real terms, define L as the limsup of the n-th roots of the absolute values: L = limsup_{n→∞} |a_n|^{1/n}. The test states: if L < 1, the series converges absolutely; if L > 1 or L = ∞, the series diverges; if L = 1, the test is inconclusive. If the limit exists instead of a limsup, the criterion simplifies to: converges when lim sqrt[n]{|a_n|} < 1, diverges when this limit is > 1, and remains inconclusive if it equals 1.

For a power series sum a_n z^n, the root test determines the radius of convergence. Let L = limsup_{n→∞} |a_n|^{1/n}. Then the radius of convergence R is R = 1/L, with the conventions that L = 0 implies R = ∞ and L = ∞ implies R = 0. The series converges absolutely for all z with |z| < R and diverges for |z| > R; on the boundary |z| = R the test is generally inconclusive and determined by other methods.

The Wurzeltest is valued for its applicability to terms without simple geometric bounds and for its straightforward criterion using root magnitudes. It is a standard tool in real and complex analysis, often used in tandem with the ratio test and other convergence criteria. The term Wurzeltest is common in German-language mathematical texts, where it is sometimes called das Wurzelkriterium.