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In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.
If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
The ratio test says the series converges if ... It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem ...
Integral test for convergence; Cauchy's convergence test; Ratio test; Direct comparison test; Limit comparison test; Root test; Alternating series test; Dirichlet's test; Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence
In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series. Integral test. The series can be ...
To prove (i) and (v), apply the ratio test and use formula above to show that whenever is not a nonnegative integer, the radius of convergence is exactly 1. Part (ii) follows from formula (), by comparison with the p-series
n November 1954, 29-year-old Sammy Davis Jr. was driving to Hollywood when a car crash left his eye mangled beyond repair. Doubting his potential as a one-eyed entertainer, the burgeoning performer sought a solution at the same venerable institution where other misfortunate starlets had gone to fill their vacant sockets: Mager & Gougelman, a family-owned business in New York City that has ...
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. ... The ratio test can be used here: