# Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series

${\displaystyle \sum _^{\infty }a_,}$

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 test

Decision diagram for the ratio test

The usual form of the test makes use of the limit

${\displaystyle L=\lim _\left|{\frac }}}\right|.}$

(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

${\displaystyle R=\lim \sup \left|{\frac }}}\right|}$
${\displaystyle r=\lim \inf \left|{\frac }}}\right|}$.

Then the ratio test states that:[2][3]

• if R < 1, the series converges absolutely;
• if r > 1, the series diverges;
• if ${\displaystyle \left|{\frac }}}\right|\geq 1}$ for all large n (regardless of the value of r), the series also diverges; this is because ${\displaystyle |a_|}$ 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.

## Examples

### Convergent because L < 1

Consider the series

${\displaystyle \sum _^{\infty }{\frac }}}$

Applying the ratio test, one computes the limit

${\displaystyle L=\lim _\left|{\frac }}}\right|=\lim _\left|{\frac {\frac }}{\frac }}}\right|={\frac }<1.}$

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

### Divergent because L > 1

Consider the series

${\displaystyle \sum _^{\infty }{\frac }}.}$

Putting this into the ratio test:

${\displaystyle L=\lim _\left|{\frac }}}\right|=\lim _\left|{\frac {\frac }}{\frac }}}\right|=e>1.}$

Thus the series diverges.

### Inconclusive because L = 1

Consider the three series

${\displaystyle \sum _^{\infty }1,}$
${\displaystyle \sum _^{\infty }{\frac }},}$
${\displaystyle \sum _^{\infty }{\frac {(-1)^}}.}$

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 ${\displaystyle \left|{\frac }}}\right|}$ of the three series are respectively ${\displaystyle 1,}$   ${\displaystyle {\frac }{(n+1)^}}}$    and   ${\displaystyle {\frac }}$. So, in all three cases, one has that the limit ${\displaystyle \lim _\left|{\frac }}}\right|}$ 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

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 ${\displaystyle L=\lim _\left|{\frac }}}\right|<1}$. 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 ${\displaystyle r={\frac }}$. Then r is strictly between L and 1, and ${\displaystyle |a_| for sufficiently large n (say, n greater than N). Hence ${\displaystyle |a_| for each n > N and i > 0, and so

${\displaystyle \sum _^{\infty }|a_|=\sum _^{\infty }\left|a_\right|<\sum _^{\infty }r^|a_|=|a_|\sum _^{\infty }r^=|a_|{\frac }<\infty .}$

That is, the series converges absolutely.

On the other hand, if L > 1, then ${\displaystyle |a_|>|a_|}$ for sufficiently large n, so that the limit of the summands is non-zero. 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 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:

${\displaystyle \sum _^{\infty }a_=\sum _^a_+\sum _^{\infty }a_}$

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 ∑an. In fact, no convergence test can fully describe the convergence properties of the series.[4][10] This is because if ∑an is convergent, a second convergent series ∑bn can be found which converges more slowly: i.e., it has the property that limn->∞ (bn/an) = ∞. Furthermore, if ∑an is divergent, a second divergent series ∑bn 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 hierarchy

Augustus De Morgan proposed a hierarchy of ratio-type tests[4][9]

The ratio test parameters (${\displaystyle \rho _}$) below all generally involve terms of the form ${\displaystyle D_a_/a_-D_}$. This term may be multiplied by ${\displaystyle a_/a_}$ to yield ${\displaystyle D_-D_a_/a_}$. 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:

${\displaystyle \rho _\equiv n\left({\frac }}}-1\right)}$

(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 ${\displaystyle \rho _\geq c}$ for all n>N.
• Diverge when ${\displaystyle \rho _\leq 1}$ for all n>N.
• Otherwise, the test is inconclusive.

For the limit version,[12] the series will:

• Converge if ${\displaystyle \lim _\rho _>1}$ (this includes the case ρ = ∞)
• Diverge if ${\displaystyle \lim _\rho _<1}$.
• 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 ${\displaystyle \liminf _\rho _>1}$
• Diverge if ${\displaystyle \limsup _\rho _<1}$
• Otherwise, the test is inconclusive.
##### Proof of Raabe's test

Defining ${\displaystyle \rho _\equiv n\left({\frac }}}-1\right)}$, we need not assume the limit exists; if ${\displaystyle \limsup \rho _<1}$, then ${\displaystyle \sum a_}$ diverges, while if ${\displaystyle \liminf \rho _>1}$ the sum converges.

The proof proceeds essentially by comparison with ${\displaystyle \sum 1/n^}$. Suppose first that ${\displaystyle \limsup \rho _<1}$. Of course if ${\displaystyle \limsup \rho _<0}$ then ${\displaystyle a_\geq a_}$ for large ${\displaystyle n}$, so the sum diverges; assume then that ${\displaystyle 0\leq \limsup \rho _<1}$. There exists ${\displaystyle R<1}$ such that ${\displaystyle \rho _\leq R}$ for all ${\displaystyle n\geq N}$, which is to say that ${\displaystyle a_/a_\leq \left(1+{\frac }\right)\leq e^}$. Thus ${\displaystyle a_\geq a_e^{-R/n}}$, which implies that ${\displaystyle a_\geq a_e^{-R(1/N+\dots +1/n)}\geq ca_e^{-R\log(n)}=ca_/n^}$ for ${\displaystyle n\geq N}$; since ${\displaystyle R<1}$ this shows that ${\displaystyle \sum a_}$ 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 ${\displaystyle 1+t that was used above: Fix ${\displaystyle R}$ and ${\displaystyle N}$. Note that ${\displaystyle \log \left(1+{\frac }\right)={\frac }+O\left({\frac }}\right)}$. So ${\displaystyle \log \left(\left(1+{\frac }\right)\dots \left(1+{\frac }\right)\right)=R\left({\frac }+\dots +{\frac }\right)+O(1)=R\log(n)+O(1)}$; hence ${\displaystyle \left(1+{\frac }\right)\dots \left(1+{\frac }\right)\geq cn^}$.

Suppose now that ${\displaystyle \liminf \rho _>1}$. Arguing as in the first paragraph, using the inequality established in the previous paragraph, we see that there exists ${\displaystyle R>1}$ such that ${\displaystyle a_\leq ca_n^{-R}}$ for ${\displaystyle n\geq N}$; since ${\displaystyle R>1}$ this shows that ${\displaystyle \sum a_}$ converges.

#### 3. Bertrand’s test

This extension is due to Joseph Bertrand and Augustus De Morgan.

Defining:

${\displaystyle \rho _\equiv n\ln n\left({\frac }}}-1\right)-\ln n}$

Bertrand's test[4][10] asserts that the series will:

• Converge when there exists a c>1 such that ${\displaystyle \rho _\geq c}$ for all n>N.
• Diverge when ${\displaystyle \rho _\leq 1}$ for all n>N.
• Otherwise, the test is inconclusive.

For the limit version, the series will:

• Converge if ${\displaystyle \lim _\rho _>1}$ (this includes the case ρ = ∞)
• Diverge if ${\displaystyle \lim _\rho _<1}$.
• 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 ${\displaystyle \liminf \rho _>1}$
• Diverge if ${\displaystyle \limsup \rho _<1}$
• Otherwise, the test is inconclusive.

#### 4. Gauss’s test

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]

${\displaystyle {\frac }}}=1+{\frac {\rho }}+{\frac }}}}$

then the series will:

• Converge if ${\displaystyle \rho >1}$
• Diverge if ${\displaystyle \rho \leq 1}$

#### 5. Kummer’s test

This extension is due to Ernst Kummer.

Let ζn be an auxiliary sequence of positive constants. Define

${\displaystyle \rho _\equiv \left(\zeta _{\frac }}}-\zeta _\right)}$

Kummer's test states that the series will:[5][6][10][11]

• Converge if there exists a ${\displaystyle c>0}$ such that ${\displaystyle \rho _\geq c}$ for all n>N. (Note this is not the same as saying ${\displaystyle \rho _>0}$)
• Diverge if ${\displaystyle \rho _\leq 0}$ for all n>N and ${\displaystyle \sum _^{\infty }1/\zeta _}$ diverges.

For the limit version, the series will:[14][7][9]

• Converge if ${\displaystyle \lim _\rho _>0}$ (this includes the case ρ = ∞)
• Diverge if ${\displaystyle \lim _\rho _<0}$ and ${\displaystyle \sum _^{\infty }1/\zeta _}$ 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 ${\displaystyle \liminf _\rho _>0}$
• Diverge if ${\displaystyle \limsup _\rho _<0}$ and ${\displaystyle \sum 1/\zeta _}$ 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 ζn=1. Then:
${\displaystyle \rho _=\left({\frac }}}-1\right)=1/\rho _-1}$
• For Raabe's test, let ζn=n. Then:
${\displaystyle \rho _=\left(n{\frac }}}-(n+1)\right)=\rho _-1}$
• For Bertrand's test, let ζn=n ln(n). Then:
${\displaystyle \rho _=n\ln(n)\left({\frac }}}-1\right)-(n+1)\ln(n+1)}$
Using ${\displaystyle \ln(n+1)=\ln(n)+\ln(1+1/n)}$ and approximating ${\displaystyle \ln(1+1/n)\rightarrow 1/n}$ for large n, which is negligible compared to the other terms, ρKummer may be written:
${\displaystyle \rho _=n\ln(n)\left({\frac }}}-1\right)-\ln(n)-1=\rho _-1}$

Note that for these three tests, the higher they are in the De Morgan hierarchy, the more slowly the 1/ζn series diverges.

##### Proof of Kummer's test

If ${\displaystyle \rho _>0}$ then fix a positive number ${\displaystyle 0<\delta <\rho _}$. There exists a natural number ${\displaystyle N}$ such that for every ${\displaystyle n>N,}$

${\displaystyle \delta \leq \zeta _{\frac }}}-\zeta _.}$

Since ${\displaystyle a_>0}$, for every ${\displaystyle n>N,}$

${\displaystyle 0\leq \delta a_\leq \zeta _a_-\zeta _a_.}$

In particular ${\displaystyle \zeta _a_\leq \zeta _a_}$ for all ${\displaystyle n\geq N}$ which means that starting from the index ${\displaystyle N}$ the sequence ${\displaystyle \zeta _a_>0}$ is monotonically decreasing and positive which in particular implies that it is bounded below by 0. Therefore, the limit

${\displaystyle \lim _\zeta _a_=L}$ exists.

This implies that the positive telescoping series

${\displaystyle \sum _^{\infty }\left(\zeta _a_-\zeta _a_\right)}$ is convergent,

and since for all ${\displaystyle n>N,}$

${\displaystyle \delta a_\leq \zeta _a_-\zeta _a_}$

by the direct comparison test for positive series, the series ${\displaystyle \sum _^{\infty }\delta a_}$ is convergent.

On the other hand, if ${\displaystyle \rho <0}$, then there is an N such that ${\displaystyle \zeta _a_}$ is increasing for ${\displaystyle n>N}$. In particular, there exists an ${\displaystyle \epsilon >0}$ for which ${\displaystyle \zeta _a_>\epsilon }$ for all ${\displaystyle n>N}$, and so ${\displaystyle \sum _a_=\sum _{\frac \zeta _}{\zeta _}}}$ diverges by comparison with ${\displaystyle \sum _{\frac {\epsilon }{\zeta _}}}$.

### Second ratio test

A more refined ratio test is the second ratio test:[7][9] For ${\displaystyle a_>0}$ define:

 ${\displaystyle L_\equiv \lim _{\frac }}}}$ ${\displaystyle L_\equiv \lim _{\frac }}}}$ ${\displaystyle L\equiv \max(L_,L_)}$

By the second ratio test, the series will:

• Converge if ${\displaystyle L<{\frac }}$
• Diverge if ${\displaystyle L>{\frac }}$
• If ${\displaystyle L={\frac }}$ then the test is inconclusive.

If the above limits do not exist, it may be possible to use the limits superior and inferior. Define:

 ${\displaystyle L_\equiv \limsup _{\frac }}}}$ ${\displaystyle L_\equiv \limsup _{\frac }}}}$ ${\displaystyle \ell _\equiv \liminf _{\frac }}}}$ ${\displaystyle \ell _\equiv \liminf _{\frac }}}}$ ${\displaystyle L\equiv \max(L_,L_)}$ ${\displaystyle \ell \equiv \min(\ell _,\ell _)}$

Then the series will:

• Converge if ${\displaystyle L<{\frac }}$
• Diverge if ${\displaystyle \ell >{\frac }}$
• If ${\displaystyle \ell \leq {\frac }\leq L}$ then the test is inconclusive.

The second ratio test can be generalized to an m-th ratio test, but higher orders are not found to be as useful.[7][9]

## Footnotes

1. ^
2. ^ Rudin 1976, §3.34
3. ^ Apostol 1974, §8.14
4. 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 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. 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.
8. ^ Samelson, Hans (November 1995). "More on Kummer's Test". The American Mathematical Monthly. 102 (9): 817–818. doi:10.2307/2974510. JSTOR 2974510.
9. 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. 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 Ďuriš, František (2 February 2018). "On Kummer's test of convergence and its relation to basic comparison tests". arXiv:1612.05167 [math.HO].
12. ^
13. ^
14. ^