Root test

The root test determines whether an infinite series converges absolutely or diverges by examining the limiting nth root of the absolute value of its terms.

Cauchy root test: A series convergence test that uses the limit superior of the nth roots of its terms. This formulation handles sequences whose nth-root values do not converge.
Infinite series: The sum of the terms of an infinite sequence, defined through limits of partial sums. The root test classifies convergence of these sums.
Power series: An infinite series whose terms are powers of a variable multiplied by fixed coefficients. The root test is a standard way to find where a power series converges.
Comparison test: A series test that compares nonnegative terms with those of a known series. Unlike direct term comparison, the root test extracts an exponential rate from each term.
Geometric series: A series whose successive terms have a constant ratio. The root test compares term magnitudes with a geometric series.
Sequence limit: A value that sequence terms approach as their index tends to infinity. The basic test uses a limit of nth-root values.
Taylor series: A power series representing a function through its derivatives at a chosen point. Root-test calculations can determine the interval of convergence of Taylor expansions.
Integral test: A convergence test relating a positive decreasing series to an improper integral. It suits smooth decreasing terms, while the root test targets nth-power structure.
Absolute convergence: Convergence of a series formed by taking the absolute value of every term. A root-test limit below one proves absolute convergence.
Absolute value: A number's nonnegative distance from zero on the real line. The test applies nth roots to term magnitudes, regardless of sign.

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