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Concept
Ratio test
Summary
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 an 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.
The usual form of the test makes use of the limit
The ratio test states that:
if L < 1 then the series converges absolutely;
if L > 1 then the series diverges;
if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let
Then the ratio test states that:
if R < 1, the series converges absolutely;
if r > 1, the series diverges;
if for all large n (regardless of the value of r), the series also diverges; this is because is nonzero and increasing and hence an does not approach zero;
the test is otherwise inconclusive.
If the limit L in () exists, we must have L = R = r. So the original ratio test is a weaker version of the refined one.
Consider the series
Applying the ratio test, one computes the limit
Since this limit is less than 1, the series converges.
Consider the series
Putting this into the ratio test:
Thus the series diverges.
Consider the three series
The first series (1 + 1 + 1 + 1 + ⋯) diverges, the second one (the one central to the Basel problem) converges absolutely and the third one (the alternating harmonic series) converges conditionally. However, the term-by-term magnitude ratios of the three series are respectively and . So, in all three cases, one has that the limit is equal to 1. This illustrates that when L = 1, the series may converge or diverge, and hence the original ratio test is inconclusive. In such cases, more refined tests are required to determine convergence or divergence.
Official source
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