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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.
Given the following geometric series:
of any two adjacent terms is 1/2. The sum of the first m terms is given by:
As m increases, this converges to 1, so the sum of the series is 1. On the other hand given this geometric series:
The quotient of any two adjacent terms is 2. The sum of the first m terms is given by
which increases without bound as m increases, so this series diverges. More generally, the sum of the first m terms of the geometric series is given by:
Whether this converges or diverges as m increases depends on whether r, the quotient of any two adjacent terms, is less than or greater than 1. Now consider the series:
This is similar to the first convergent sequence above, except that now the ratio of two terms is not fixed at exactly 1/2:
However, as n increases, the ratio still tends in the limit towards the same constant 1/2. The ratio test generalizes the simple test for geometric series to more complex series like this one where the quotient of two terms is not fixed, but in the limit tends towards a fixed value. The rules are similar: if the quotient approaches a value less than one, the series converges, whereas if it approaches a value greater than one, the series diverges.
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 does not converge;
- 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
- 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 (1) exists, we must have L=R=r. So the original ratio test is a weaker version of the refined one.
Convergent because L<1
Consider the series or sequence of series
Putting this into the ratio test:
As every term is positive, the series converges.
Divergent because L>1
Consider the series
Putting this into the ratio test:
Thus the series diverges.
Inconclusive because L=1
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, we have. This illustrates that when L=1, the series may converge or diverge and hence the original ratio test is inconclusive. For the first series , however, as the term-by-term magnitude ratio for all n, we can apply the third criterion in the refined version of the ratio test to conclude that the series diverges.
Below is a proof of the validity of the original ratio test.
Suppose that . We can then show that the series converges absolutely by showing that its terms will eventually become less than those of a certain convergent geometric series. To do this, let . Then r is strictly between L and 1, and for sufficiently large n (say, n greater than N). Hence for each n > N and i > 0, and so
That is, the series converges absolutely.
On the other hand, if L > 1, then for sufficiently large n, so that the limit of the summands is non-zero. Hence the series diverges.
Extensions for L = 1
|This section needs expansion with: De Morgan's hierarchy and usage examples. And proofs should be provided.. You can help by adding to it. (August 2013)|
As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to ratio test, however, sometimes allows one to deal with this case. For instance, the aforementioned refined version of the test handles the case
Below are some other extensions.
This extension is due to Joseph Ludwig Raabe. It states that if
then the series will be absolutely convergent if R>1 and divergent if R<1. d'Alembert's ratio test and Raabe's test are the first and second theorems in a hierarchy of such theorems due to Augustus De Morgan.
Higher order tests
The next cases in de Morgan's hierarchy are Bertrand's and Gauss's test. Each test involves slightly different higher order asymptotics. Bertrand's test asserts that if
then the series converges if lim inf ρn > 1, and diverges if lim sup ρn < 1.
Gauss's test asserts that if
where r > 1 and Cn is bounded, then the series converges if h > 1 and diverges if h ≤ 1.
These are both special cases of Kummer's test for the convergence of the series Σan, for positive an. Let ζn be an auxiliary sequence of positive constants. Let
Then if ρ > 0, the series converges. If ρ < 0 and Σ1/ζn diverges, then the series diverges. Otherwise the test is inconclusive.
Proof of Kummer's test
If then fix a positive number . There exists a natural number such that for every
Since , for every
In particular for all which means that starting from the index the sequence is monotonically decreasing and positive which in particular implies that it is bounded below by 0. Therefore the limit
This implies that the positive telescoping series
- is convergent,
and since for all
by the direct comparison test for positive series, the series is convergent.
On the other hand, if , then there is an N such that is increasing for . In particular, there exists an for which for all , and so diverges by comparison with .
- d'Alembert, J. (1768), Opuscules, V, pp. 171–183.
- Apostol, Tom M. (1974), Mathematical analysis (2nd ed.), Addison-Wesley, ISBN 978-0-201-00288-1: §8.14.
- Knopp, Konrad (1956), Infinite Sequences and Series, New York: Dover publications, Inc., ISBN 0-486-60153-6: §3.3, 5.4.
- Rudin, Walter (1976), Principles of Mathematical Analysis (3rd ed.), New York: McGraw-Hill, Inc., ISBN 0-07-054235-X: §3.34.
- Hazewinkel, Michiel, ed. (2001), "Bertrand criterion", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Hazewinkel, Michiel, ed. (2001), "Gauss criterion", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Hazewinkel, Michiel, ed. (2001), "Kummer criterion", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Watson, G. N.; Whittaker, E. T. (1963), A Course in Modern Analysis (4th ed.), Cambridge University Press, ISBN 0-521-58807-3: §2.36, 2.37.