Ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
Part of a series of articles about  
Calculus  





Specialized 

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 test
The usual form of the test makes use of the limit

(1)
The ratio test states that:
 if L < 1 then the series converges absolutely;
 if L > 1 then the series is divergent;
 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:[2][3]
 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 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
Applying the ratio test, one computes the limit
Since this limit is less than 1, 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 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.
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 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.
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 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 ∑a_{n}. In fact, no convergence test can fully describe the convergence properties of the series.[4][10] This is because if ∑a_{n} is convergent, a second convergent series ∑b_{n} can be found which converges more slowly: i.e., it has the property that lim_{n>∞} (b_{n}/a_{n}) = ∞. Furthermore, if ∑a_{n} is divergent, a second divergent series ∑b_{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.
De Morgan hierarchy
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.
1. d’Alembert’s ratio test
The first test in the De Morgan hierarchy is the ratio test as described above.
2. Raabe's test
This extension is due to Joseph Ludwig Raabe. Define:
(and some extra terms, see Ali, Blackburn, Feld, Duris (none), Duris2)
 Converge when there exists a c>1 such that for all n>N.
 Diverge when for all n>N.
 Otherwise, the test is inconclusive.
For the limit version,[12] the series will:
 Converge if (this includes the case ρ = ∞)
 Diverge if .
 If ρ = 1, the test is inconclusive.
When the above limit does not exist, it may be possible to use limits superior and inferior.[4] The series will:
 Converge if
 Diverge if
 Otherwise, the test is inconclusive.
Proof of Raabe's test
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.
3. Bertrand’s test
This extension is due to Joseph Bertrand and Augustus De Morgan.
Defining:
Bertrand's test[4][10] asserts that the series will:
 Converge when there exists a c>1 such that for all n>N.
 Diverge when for all n>N.
 Otherwise, the test is inconclusive.
For the limit version, the series will:
 Converge if (this includes the case ρ = ∞)
 Diverge if .
 If ρ = 1, the test is inconclusive.
When the above limit does not exist, it may be possible to use limits superior and inferior.[4][9][13] The series will:
 Converge if
 Diverge if
 Otherwise, the test is inconclusive.
4. Extended Bertrand’s test
This extension appeared at the first time in [10]. A short proof that is free of technical assumptions is provided in [14].
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
 Converge when there exists a such that for all .
 Diverge when for all .
 Otherwise, the test is inconclusive.
For the limit version, the series
 Converge if (this includes the case )
 Diverge if .
 If , the test is inconclusive.
When the above limit does not exist, it may be possible to use limits superior and inferior. The series
 Converge if
 Diverge if
 Otherwise, the test is inconclusive.
For applications of Extended Bertrand's test see Birthdeath process.
5. Gauss’s test
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:
 Converge if
 Diverge if
6. Kummer’s test
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]
 Converge if there exists a such that for all n>N. (Note this is not the same as saying )
 Diverge if for all n>N and diverges.
For the limit version, the series will:[15][7][9]
 Converge if (this includes the case ρ = ∞)
 Diverge if and diverges.
 Otherwise the test is inconclusive
When the above limit does not exist, it may be possible to use limits superior and inferior.[4] The series will
 Converge if
 Diverge if and diverges.
Special cases
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]
 For the ratio test, let ζ_{n}=1. Then:
 For Raabe's test, let ζ_{n}=n. Then:
 For Bertrand's test, let ζ_{n}=n ln(n). Then:
 Using and approximating for large n, which is negligible compared to the other terms, may be written:
 For Extended Bertrand's test, let From the Taylor series expansion for large we arrive at the approximation
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.
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
 exists.
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 .
Second ratio test
A more refined ratio test is the second ratio test:[7][9] For define:
By the second ratio test, the series will:
 Converge if
 Diverge if
 If then the test is inconclusive.
If the above limits do not exist, it may be possible to use the limits superior and inferior. Define:
Then the series will:
 Converge if
 Diverge if
 If then the test is inconclusive.
The second ratio test can be generalized to an mth ratio test, but higher orders are not found to be as useful.[7][9]
See also
Footnotes
 Weisstein, Eric W. "Ratio Test". MathWorld.
 Rudin 1976, §3.34
 Apostol 1974, §8.14
 Bromwich, T. J. I’A (1908). An Introduction To The Theory of Infinite Series. Merchant Books.
 Knopp, Konrad (1954). Theory and Application of Infinite Series. London: Blackie & Son Ltd.
 Tong, Jingcheng (May 1994). "Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series". The American Mathematical Monthly. 101 (5): 450–452. doi:10.2307/2974907. JSTOR 2974907.
 Ali, Sayel A. (2008). "The mth Ratio Test: New Convergence Test for Series" (PDF). The American Mathematical Monthly. 115 (6): 514–524. doi:10.1080/00029890.2008.11920558. Retrieved 21 November 2018.
 Samelson, Hans (November 1995). "More on Kummer's Test". The American Mathematical Monthly. 102 (9): 817–818. doi:10.2307/2974510. JSTOR 2974510.
 Blackburn, Kyle (4 May 2012). "The mth Ratio Convergence Test and Other Unconventional Convergence Tests" (PDF). University of Washington College of Arts and Sciences. Retrieved 27 November 2018.
 Ďuriš, František (2009). Infinite series: Convergence tests (Bachelor's thesis). Katedra Informatiky, Fakulta Matematiky, Fyziky a Informatiky, Univerzita Komenského, Bratislava. Retrieved 28 November 2018.
 Ďuriš, František (2 February 2018). "On Kummer's test of convergence and its relation to basic comparison tests". arXiv:1612.05167 [math.HO].
 Weisstein, Eric W. "Raabe's Test". MathWorld.
 Weisstein, Eric W. "Bertrand's Test". MathWorld.
 Abramov, Vyacheslav M. "Extension of the BertrandDe Morgan test and its application" (PDF). The American Mathematical Monthly (to appear). Retrieved 28 November 2019.
 Weisstein, Eric W. "Kummer's Test". MathWorld.
References
 d'Alembert, J. (1768), Opuscules, V, pp. 171–183.
 Apostol, Tom M. (1974), Mathematical analysis (2nd ed.), AddisonWesley, ISBN 9780201002881: §8.14.
 Knopp, Konrad (1956), Infinite Sequences and Series, New York: Dover Publications, Bibcode:1956iss..book.....K, ISBN 9780486601533: §3.3, 5.4.
 Rudin, Walter (1976), Principles of Mathematical Analysis (3rd ed.), New York: McGrawHill, Inc., ISBN 9780070542358: §3.34.
 Hazewinkel, Michiel, ed. (2001) [1994], "Bertrand criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Hazewinkel, Michiel, ed. (2001) [1994], "Gauss criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Hazewinkel, Michiel, ed. (2001) [1994], "Kummer criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Watson, G. N.; Whittaker, E. T. (1963), A Course in Modern Analysis (4th ed.), Cambridge University Press, ISBN 9780521588072: §2.36, 2.37.