 Ratio test

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.
Contents
The 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 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
 and .
Then the ratio test states that:^{[1]}^{[2]}
 if R < 1, the series converges absolutely;
 if r > 1, the series diverges;
 if for all large n (regardless of whether the ratios has or has not a limit), the series also diverges; this is because  a_{n}  is nonzero and increasing and hence a_{n} 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.
Examples
Convergent because L<1
Consider the series
Putting this into the ratio test:
Thus 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
 and .
The first series diverges, the second one converges absolutely and the third one converges conditionally. However, the termbyterm magnitude ratios of the three series are respectively 1, 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 termbyterm 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.
Proof
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  a_{n + 1}  < r  a_{n}  for sufficiently large n (say, n greater than N). Hence  a_{n + i}  < r^{i}  a_{n}  for each n > N and i > 0, and so
That is, the series converges absolutely.
On the other hand, if L > 1, then  a_{n + 1}  >  a_{n}  for sufficiently large n, so that the limit of the summands is nonzero. Hence the series diverges.
Extensions for L=1
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.
Raabe's test
This extension is due to Joseph Ludwig Raabe. It states that if
and if
then the series will be absolutely convergent. d'Alembert's ratio test and Raabe's test are the first and second theorem 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. If
then the series converges if lim inf ρ_{n} > 1, and diverges if lim sup ρ_{n} < 1. This is Bertrand's test.
If
where r > 1 and C_{n} is bounded, then the series converges if h > 1 and diverges if h ≤ 1. This is Gauss's test.
These are both special cases of Kummer's test for the convergence of the series Σa_{n}. Let ζ_{n} be an auxiliary sequence of positive constants. Let
Then if ρ > 0, the series converges. If ρ < 0 and Σ1/ζ_{n} diverges, then the series diverges. Otherwise the test is inconclusive.
See also
Footnotes
 ^ Rudin 1976, §3.34
 ^ Apostol 1974, §8.14
References
 d'Alembert, J. (1768), Opuscules, V, pp. 171–183, http://gallica.bnf.fr/ark:/12148/bpt6k62424s.image.f192.
 Apostol, Tom M. (1974), Mathematical analysis (2nd ed.), AddisonWesley, ISBN 9780201002881: §8.14.
 Knopp, Konrad (1956), Infinite Sequences and Series, New York: Dover publications, Inc., ISBN 0486601536: §3.3, 5.4.
 Rudin, Walter (1976), Principles of Mathematical Analysis (3rd ed.), New York: McGrawHill, Inc., ISBN 007054235X: §3.34.
 Weisstein, Eric W., "Bertrand's Test" from MathWorld.
 Weisstein, Eric W., "Gauss's Test" from MathWorld.
 Weisstein, Eric W., "Kummer's Test" from MathWorld.
 Watson, G. N.; Whittaker, E. T. (1963), A Course in Modern Analysis (4th ed.), Cambridge University Press, ISBN 0521588073: §2.36, 2.37.
Categories: Mathematical series
 Convergence tests

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