Part of a series of articles about 
Calculus 

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.^{[1]}
The usual form of the test makes use of the limit

(1) 
The ratio test states that:
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:^{[2]}^{[3]}
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.
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 termbyterm 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.
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, consider a real number r such that . This implies that for sufficiently large n; say, for all 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 nonzero. Hence the series diverges.
As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}
In all the tests below one assumes that Σa_{n} is a sum with positive a_{n}. These tests also may be applied to any series with a finite number of negative terms. Any such series may be written as:
where a_{N} is the highestindexed negative term. The first expression on the right is a partial sum which will be finite, and so the convergence of the entire series will be determined by the convergence properties of the second expression on the right, which may be reindexed to form a series of all positive terms beginning at n=1.
Each test defines a test parameter (ρ_{n}) which specifies the behavior of that parameter needed to establish convergence or divergence. For each test, a weaker form of the test exists which will instead place restrictions upon lim_{n>∞}ρ_{n}.
All of the tests have regions in which they fail to describe the convergence properties of Σa_{n}. In fact, no convergence test can fully describe the convergence properties of the series.^{[4]}^{[10]} This is because if Σa_{n} is convergent, a second convergent series Σb_{n} can be found which converges more slowly: i.e., it has the property that lim_{n>∞} (b_{n}/a_{n}) = ∞. Furthermore, if Σa_{n} is divergent, a second divergent series Σb_{n} can be found which diverges more slowly: i.e., it has the property that lim_{n>∞} (b_{n}/a_{n}) = 0. Convergence tests essentially use the comparison test on some particular family of a_{n}, and fail for sequences which converge or diverge more slowly.
Augustus De Morgan proposed a hierarchy of ratiotype tests^{[4]}^{[9]}
The ratio test parameters () below all generally involve terms of the form . This term may be multiplied by to yield . This term can replace the former term in the definition of the test parameters and the conclusions drawn will remain the same. Accordingly, there will be no distinction drawn between references which use one or the other form of the test parameter.
The first test in the De Morgan hierarchy is the ratio test as described above.
This extension is due to Joseph Ludwig Raabe. Define:
(and some extra terms, see Ali, Blackburn, Feld, Duris (none), Duris2)
The series will:^{[7]}^{[10]}^{[9]}
For the limit version,^{[12]} the series will:
When the above limit does not exist, it may be possible to use limits superior and inferior.^{[4]} The series will:
Defining , we need not assume the limit exists; if , then diverges, while if the sum converges.
The proof proceeds essentially by comparison with . Suppose first that . Of course if then for large , so the sum diverges; assume then that . There exists such that for all , which is to say that . Thus , which implies that for ; since this shows that diverges.
The proof of the other half is entirely analogous, with most of the inequalities simply reversed. We need a preliminary inequality to use in place of the simple that was used above: Fix and . Note that . So ; hence .
Suppose now that . Arguing as in the first paragraph, using the inequality established in the previous paragraph, we see that there exists such that for ; since this shows that converges.
This extension is due to Joseph Bertrand and Augustus De Morgan.
Defining:
Bertrand's test^{[4]}^{[10]} asserts that the series will:
For the limit version, the series will:
When the above limit does not exist, it may be possible to use limits superior and inferior.^{[4]}^{[9]}^{[13]} The series will:
This extension probably appeared at the first time by Margaret Martin in.^{[14]} A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) is provided in.^{[15]}
Let be an integer, and let denote the th iterate of natural logarithm, i.e. and for any , .
Suppose that the ratio , when is large, can be presented in the form
(The empty sum is assumed to be 0. With , the test reduces to Bertrand's test.)
The value can be presented explicitly in the form
Extended Bertrand's test asserts that the series
For the limit version, the series
When the above limit does not exist, it may be possible to use limits superior and inferior. The series
For applications of Extended Bertrand's test see birth–death process.
This extension is due to Carl Friedrich Gauss.
Assuming a_{n} > 0 and r > 1, if a bounded sequence C_{n} can be found such that for all n:^{[5]}^{[7]}^{[9]}^{[10]}
then the series will:
This extension is due to Ernst Kummer.
Let ζ_{n} be an auxiliary sequence of positive constants. Define
Kummer's test states that the series will:^{[5]}^{[6]}^{[10]}^{[11]}
For the limit version, the series will:^{[16]}^{[7]}^{[9]}
When the above limit does not exist, it may be possible to use limits superior and inferior.^{[4]} The series will
All of the tests in De Morgan's hierarchy except Gauss's test can easily be seen as special cases of Kummer's test:^{[4]}
where the empty product is assumed to be 1. Then,
Hence,
Note that for these four tests, the higher they are in the De Morgan hierarchy, the more slowly the series diverges.
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
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 .
A new version of Kummer's test was established by Tong.^{[6]} See also ^{[8]} ^{[11]}^{[17]} for further discussions and new proofs. The provided modification of Kummer's theorem characterizes all positive series, and the convergence or divergence can be formulated in the form of two necessary and sufficient conditions, one for convergence and another for divergence.
A more refined ratio test is the second ratio test:^{[7]}^{[9]} For define:
By the second ratio test, the series will:
If the above limits do not exist, it may be possible to use the limits superior and inferior. Define:
Then the series will:
This test is a direct extension of the second ratio test.^{[7]}^{[9]} For and positive define:
By the th ratio test, the series will:
If the above limits do not exist, it may be possible to use the limits superior and inferior. For define:
Then the series will:
This test is an extension of the th ratio test.^{[18]}
Assume that the sequence is a positive decreasing sequence.
Let be such that exists. Denote , and assume .
Assume also that
Then the series will: