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

where each term is a real or complex number and all but finitely many terms are non-zero. 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.

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 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

1. .

Then the ratio test states that:

• if R < 1, the series converges absolutely;

• if r for all large n (regardless of the value of r is nonzero and increasing and hence does not approach zero;

• the test is otherwise inconclusive.

If the limit L in () 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 (the one central to the Basel problem) converges absolutely and the third (the alternating harmonic series and is equal to 1. This illustrates that when L = 1, the series may converge or diverge: the 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 generalized ratio test.

is the limit inferior diverges by the n-th term test. is the geometric series converges by the monotone convergence theorem converges by the absolute convergence test., this gives the original ratio test.

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 allow one to deal with this case.

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

.

.

. In fact, no convergence test can fully describe the convergence properties of the series., and fail for sequences which converge or diverge more slowly.

De Morgan hierarchy

Augustus De Morgan proposed a hierarchy of ratio-type tests

. 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)

The series will:

• .

• .

• Otherwise, the test is inconclusive.

For the limit version, the series will:

• )

• .

• , the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior. The series will:

• Otherwise, the test is inconclusive.

Proof of Raabe's test

the sum converges.

diverges.

.

converges.

is diverges.)

3. Bertrand's test

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

Defining:

Bertrand's test asserts that the series will:

• .

• .

• Otherwise, the test is inconclusive.

For the limit version, the series will:

• )

• .

• , the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior. The series will:

• Otherwise, the test is inconclusive.

4. Extended Bertrand's test

This extension probably appeared at the first time by Margaret Martin in 1941. A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) was provided by Vyacheslav Abramov in 2019.

th iterate of natural logarithm.

is large, can be presented in the form

, the test reduces to Bertrand's test.)

can be presented explicitly in the form

Extended Bertrand's test asserts that the series

• .

• .

• Otherwise, the test is inconclusive.

For the limit version, the series

• )

• .

• , the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior. The series

• Otherwise, the test is inconclusive.

For applications of Extended Bertrand's test see birth–death process.

5. Gauss's test

This extension is due to Carl Friedrich Gauss.

:

then the series will:

6. Kummer's test

This extension is due to Ernst Kummer.

be an auxiliary sequence of positive constants. Define

Kummer's test states that the series will:

• )

• diverges.

For the limit version, the series will:

• )

• diverges.

• Otherwise the test is inconclusive

When the above limit does not exist, it may be possible to use limits superior and inferior. The series will

• 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:

• . Then:

  . Then:

  . Then:

• and approximating for large n may be written:

• . From the Taylor series we arrive at the approximation

  1. ,

where the empty product is assumed to be 1. Then,

1. .

Hence,

1. .

series diverges.

Proof of Kummer's test

,

1. .

,

1. .

is monotonically decreasing and positive which in particular implies that it is bounded below by 0. Therefore, the limit

1. exists.

This implies that the positive telescoping series

1. is convergent,

,

by the direct comparison test is convergent.

, then there is an N.

Tong's modification of Kummer's test

A new version of Kummer's test was established by Tong. See also 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.

The first of these statements can be simplified as follows:

The second statement can be simplified similarly:

Frink's ratio test

Another ratio test that can be set in the framework of Kummer's theorem was presented by Orrin Frink 1948.

,

• converges absolutely.

• diverges.

with a power series, and can be seen to be related to Raabe's test.

Ali's second ratio test

A more refined ratio test is the second ratio test: define:By the second ratio test, the series will:

• 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:

• then the test is inconclusive.

Ali's mth ratio test

This test is a direct extension of the second ratio test. define:th ratio test, the series will:

• then the test is inconclusive.

define:Then the series will:

• , then the test is inconclusive.

Ali--Deutsche Cohen φ-ratio test

th ratio test.

is a positive decreasing sequence.

.

Then the series will:

• , then the test is inconclusive.

See also

• Root test

• Radius of convergence

Footnotes

References

• .

• : §8.14.

• : §3.3, 5.4.

• : §3.34.

• : §2.36, 2.37.

it:Criteri di convergenza#Criterio del rapporto (o di d'Alembert)