Wurzelkriterium

The Wurzelkriterium, or root test, is a mathematical test used to determine the convergence or divergence of an infinite series. It is particularly useful for series where terms involve powers or factorials. The test examines the limit of the nth root of the absolute value of the nth term of the series.

Let the series be denoted by the sum of a_n from n=1 to infinity. The Wurzelkriterium states that if the limit of the absolute value of the nth root of a_n as n approaches infinity is less than 1, then the series converges absolutely. If this limit is greater than 1, or if it is infinity, then the series diverges. If the limit is exactly 1, the test is inconclusive, and other convergence tests must be employed.

The formula for the Wurzelkriterium is as follows: Let L = lim (n->infinity) |a_n|^(1/n). If L < 1, the series converges absolutely. If L > 1 or L = infinity, the series diverges. If L = 1, the test provides no information about the convergence of the series. This test is a powerful tool in calculus for analyzing the behavior of infinite sums.