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.[1]

The testEdit

 
Decision diagram for the ratio 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 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:[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 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.

ExamplesEdit

Convergent because L < 1Edit

Consider the series

 

Applying the ratio test, one computes the limit

 

Since this limit is less than 1, the series converges.

Divergent because L > 1Edit

Consider the series

 

Putting this into the ratio test:

 

Thus the series diverges.

Inconclusive because L = 1Edit

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.

ProofEdit

 
In this example, the ratio of adjacent terms in the blue sequence converges to L=1/2. We choose r = (L+1)/2 = 3/4. Then the blue sequence is dominated by the red sequence rk for all n ≥ 2. The red sequence converges, so the blue sequence does as well.

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 non-zero. Hence the series diverges.

Extensions for L = 1Edit

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 Σan is a sum with positive an. These tests also may be applied to any series with a finite number of negative terms. Any such series may be written as:

 

where aN is the highest-indexed 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 re-indexed to form a series of all positive terms beginning at n=1.

Each test defines a test parameter (ρn) 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 limn->∞ρn.

All of the tests have regions in which they fail to describe the convergence properties of Σan. In fact, no convergence test can fully describe the convergence properties of the series.[4][10] This is because if Σan is convergent, a second convergent series Σbn can be found which converges more slowly: i.e., it has the property that limn->∞ (bn/an) = ∞. Furthermore, if Σan is divergent, a second divergent series Σbn can be found which diverges more slowly: i.e., it has the property that limn->∞ (bn/an) = 0. Convergence tests essentially use the comparison test on some particular family of an, and fail for sequences which converge or diverge more slowly.

De Morgan hierarchyEdit

Augustus De Morgan proposed a hierarchy of ratio-type 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 testEdit

The first test in the De Morgan hierarchy is the ratio test as described above.

2. Raabe's testEdit

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]

  • 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 testEdit

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 testEdit

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 testEdit

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

  • 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 birth–death process.

5. Gauss's testEdit

This extension is due to Carl Friedrich Gauss.

Assuming an > 0 and r > 1, if a bounded sequence Cn can be found such that for all n:[5][7][9][10]

 

then the series will:

  • Converge if  
  • Diverge if  

6. Kummer's testEdit

This extension is due to Ernst Kummer.

Let ζn 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:[16][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 casesEdit

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 ζn=1. Then:
 
  • For Raabe's test, let ζn=n. Then:
 
  • For Bertrand's test, let ζn=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 testEdit

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  .

Tong's modification of Kummer's testEdit

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.

  • Series   converges if and only if there exists a positive sequence  ,  , such that  
  • Series   diverges if and only if there exists a positive sequence  ,  , such that   and  

The first of these statements can be simplified as follows: [18]

  • Series   converges if and only if there exists a positive sequence  ,  , such that  

The second statement can be simplified similarly:

  • Series   diverges if and only if there exists a positive sequence  ,  , such that   and  

However, it becomes useless, since the condition   in this case reduces to the original claim  

Ali's second ratio testEdit

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.

Ali's mth ratio testEdit

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:

  • 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. For   define:

 
 
   

Then the series will:

  • Converge if  
  • Diverge if  
  • If  , then the test is inconclusive.

Ali--Deutsche Cohen φ-ratio testEdit

This test is an extension of the  th ratio test.[19]

Assume that the sequence   is a positive decreasing sequence.

Let   be such that   exists. Denote  , and assume  .

Assume also that  

Then the series will:

  • Converge if  
  • Diverge if  
  • If  , then the test is inconclusive.

See alsoEdit

FootnotesEdit

  1. ^ Weisstein, Eric W. "Ratio Test". MathWorld.
  2. ^ Rudin 1976, §3.34
  3. ^ Apostol 1974, §8.14
  4. ^ a b c d e f g h Bromwich, T. J. I'A (1908). An Introduction To The Theory of Infinite Series. Merchant Books.
  5. ^ a b c Knopp, Konrad (1954). Theory and Application of Infinite Series. London: Blackie & Son Ltd.
  6. ^ a b c 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.
  7. ^ a b c d e f 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. S2CID 16336333. Retrieved 21 November 2018.
  8. ^ a b Samelson, Hans (November 1995). "More on Kummer's Test". The American Mathematical Monthly. 102 (9): 817–818. doi:10.2307/2974510. JSTOR 2974510.
  9. ^ a b c d e f g h 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.
  10. ^ a b c d e f Ďuriš, František (2009). Infinite series: Convergence tests (Bachelor's thesis). Katedra Informatiky, Fakulta Matematiky, Fyziky a Informatiky, Univerzita Komenského, Bratislava. Retrieved 28 November 2018.
  11. ^ a b c Ďuriš, František (2 February 2018). "On Kummer's test of convergence and its relation to basic comparison tests". arXiv:1612.05167 [math.HO].
  12. ^ Weisstein, Eric W. "Raabe's Test". MathWorld.
  13. ^ Weisstein, Eric W. "Bertrand's Test". MathWorld.
  14. ^ Martin, Margaret (1941). "A sequence of limit tests for the convergence of series" (PDF). Bulletin of the American Mathematical Society. 47 (6): 452–457. doi:10.1090/S0002-9904-1941-07477-X.
  15. ^ Abramov, Vyacheslav M. (May 2020). "Extension of the Bertrand–De Morgan test and its application". The American Mathematical Monthly. 127 (5): 444–448. arXiv:1901.05843. doi:10.1080/00029890.2020.1722551. S2CID 199552015.
  16. ^ Weisstein, Eric W. "Kummer's Test". MathWorld.
  17. ^ Abramov, Vyacheslav, M. (21 June 2021). "A simple proof of Tong's theorem". arXiv:2106.13808 [math.HO].
  18. ^ Abramov, Vyacheslav M. (May 2022). "Evaluating the sum of convergent positive series" (PDF). Publications de l'Institut Mathématique. Nouvelle Série. 111 (125): 41–53. doi:10.2298/PIM2225041A. S2CID 237499616.
  19. ^ Ali, Sayel; Cohen, Marion Deutsche (2012). "phi-ratio tests". Elemente der Mathematik. 67 (4): 164–168. doi:10.4171/EM/206.

ReferencesEdit