Ratio test

The ratio test determines whether an infinite series converges absolutely or diverges by examining the limit of the ratio of successive term magnitudes.

Limit comparison test: A series test that compares the terms of two positive-term series through the limit of their term ratios. The ratio test’s geometric comparison is a specialized comparison of successive terms.
Taylor series: A power series representing a function through its derivatives at a specified point. Term ratios can determine where a Taylor series converges.
Infinite series: A sum defined as the limit of the partial sums of an infinite sequence of terms. The ratio test decides convergence properties of this kind of sum.
Jean le Rond d'Alembert: An eighteenth-century French mathematician and philosopher who studied convergence criteria for infinite series. The ratio test is also known as d'Alembert’s ratio test.
Geometric series: A series whose successive terms have a constant ratio. When successive term ratios approach a number below one, the series behaves like a convergent geometric series.
Exponential function: The function equal to its own derivative and represented by the power series with coefficients one over factorials. Its factorial denominators make the ratio test yield convergence for every real input.
Sequence limit: A value that the terms of a sequence approach as their index grows without bound. The test’s conclusion depends on the limiting value of successive term ratios.
Augustin-Louis Cauchy: A French mathematician whose work helped establish rigorous foundations for analysis. Cauchy developed a closely related root criterion for series convergence.
Absolute convergence: Convergence of a series of terms’ absolute values, which guarantees convergence of the original series. A ratio-test limit below one establishes absolute convergence, even for signed terms.
Binomial series: The power-series expansion of a real or complex power of one plus a variable. The ratio test identifies the interior of its convergence interval.

Linked from 16 pages