Convergence tests

Short description: Mathematical criterion about whether a series converges

Part of a series of articles about

Calculus

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Series

* Geometric (arithmetico-geometric)
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Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

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Specialized

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Glossary of calculus

* Glossary of calculus

* v
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* e

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ∑n=1∞an.

## Contents

* 1 List of tests
* 1.1 Limit of the summand
* 1.2 Ratio test
* 1.3 Root test
* 1.4 Integral test
* 1.4.1 p-series test
* 1.5 Direct comparison test
* 1.6 Limit comparison test
* 1.7 Cauchy condensation test
* 1.8 Abel's test
* 1.9 Absolute convergence test
* 1.10 Alternating series test
* 1.11 Dirichlet's test
* 1.12 Cauchy's convergence test
* 1.13 Stolz–Cesàro theorem
* 1.14 Weierstrass M-test
* 1.15 Extensions to the ratio test
* 1.15.1 Raabe–Duhamel's test
* 1.15.2 Bertrand's test
* 1.15.3 Gauss's test
* 1.15.4 Kummer's test
* 1.16 Notes
* 2 Examples
* 3 Convergence of products
* 4 See also
* 5 References
* 6 Further reading

## List of tests

### Limit of the summand

If the limit of the summand is undefined or nonzero, that is limn->∞an≠0, then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.

### Ratio test

This is also known as d'Alembert's criterion.

Suppose that there exists r such that

limn->∞|an+1an|=r.
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

### Root test

This is also known as the nth root test or Cauchy's criterion.

Let

r=lim supn->∞|an|n,
where lim sup denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.[1]

### Integral test

The series can be compared to an integral to establish convergence or divergence. Let f:[1,∞)->ℝ+ be a non-negative and monotonically decreasing function such that f(n)=an. If ∫1∞f(x)dx=limt->∞∫1tf(x)dx<∞, then the series converges. But if the integral diverges, then the series does so as well. In other words, the series an converges if and only if the integral converges.

#### p-series test

A commonly-used corollary of the integral test is the p-series test. Let k>0. Then ∑n=k∞(1np) converges if p>1.

The case of p=1,k=1 yields the harmonic series, which diverges. The case of p=2,k=1 is the Basel problem and the series converges to π26. In general, for p>1,k=1, the series is equal to the Riemann zeta function applied to p, that is ζ(p).

### Direct comparison test

If the series ∑n=1∞bn is an absolutely convergent series and |an|≤|bn| for sufficiently large n , then the series ∑n=1∞an converges absolutely.

### Limit comparison test

If {an},{bn}>0, (that is, each element of the two sequences is positive) and the limit limn->∞anbn exists, is finite and non-zero, then either both series converge or both series diverge.

### Cauchy condensation test

Let {an} be a non-negative non-increasing sequence. Then the sum A=∑n=1∞an converges if and only if the sum A*=∑n=0∞2na2n converges. Moreover, if they converge, then A≤A*≤2A holds.

### Abel's test

Suppose the following statements are true:

1. ∑an is a convergent series,
2. {bn} is a monotonic sequence, and
3. {bn} is bounded.

Then ∑anbn is also convergent.

### Absolute convergence test

Every absolutely convergent series converges.

### Alternating series test

Suppose the following statements are true:

* an are all positive,
* limn->∞an=0 and
* for every n, an+1≤an.

Then ∑n=1∞(−1)nan and ∑n=1∞(−1)n+1an are convergent series. This test is also known as the Leibniz criterion.

### Dirichlet's test

If {an} is a sequence of real numbers and {bn} a sequence of complex numbers satisfying

* an≥an+1

* limn->∞an=0

* |∑n=1Nbn|≤M for every positive integer N

where M is some constant, then the series

∑n=1∞anbn

converges.

### Cauchy's convergence test

A series ∑i=0∞ai is convergent if and only if for every ε>0 there is a natural number N such that

|an+1+an+2+⋯+an+p|<ε

holds for all n > N and all p ≥ 1.

### Stolz–Cesàro theorem

Let (an)n≥1 and (bn)n≥1 be two sequences of real numbers. Assume that (bn)n≥1 is a strictly monotone and divergent sequence and the following limit exists:

limn->∞an+1−anbn+1−bn=l.

Then, the limit

limn->∞anbn=l.

### Weierstrass M-test

Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions

* |fn(x)|≤Mn for all n≥1 and all x∈A, and
* ∑n=1∞Mn converges.

Then the series

∑n=1∞fn(x)

converges absolutely and uniformly on A.

### Extensions to the ratio test

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.

#### Raabe–Duhamel's test

Let { an } be a sequence of positive numbers.

Define

bn=n(anan+1−1).

L=limn->∞bn

exists there are three possibilities:

* if L > 1 the series converges (this includes the case L = ∞)
* if L < 1 the series diverges
* and if L = 1 the test is inconclusive.

An alternative formulation of this test is as follows. Let { an } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that

|an+1an|≤1−bn

for all n > K then the series {an} is convergent.

#### Bertrand's test

Let { an } be a sequence of positive numbers.

Define

bn=ln⁡n(n(anan+1−1)−1).

L=limn->∞bn

exists, there are three possibilities:[2][3]

* if L > 1 the series converges (this includes the case L = ∞)
* if L < 1 the series diverges
* and if L = 1 the test is inconclusive.

#### Gauss's test

Let { an } be a sequence of positive numbers. If anan+1=1+αn+O(1/nβ) for some β > 1, then ∑an converges if α > 1 and diverges if α ≤ 1.[4]

#### Kummer's test

Let { an } be a sequence of positive numbers. Then:[5][6][7]

(1) ∑an converges if and only if there is a sequence bn of positive numbers and a real number c > 0 such that bk(ak/ak+1)−bk+1≥c.

(2) ∑an diverges if and only if there is a sequence bn of positive numbers such that bk(ak/ak+1)−bk+1≤0

and ∑1/bn diverges.

### Notes

* For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

## Examples

Consider the series

∑n=1∞1nα.

|
|

(i)

Cauchy condensation test implies that (i) is finitely convergent if

∑n=1∞2n(12n)α

|
|

(ii)

is finitely convergent. Since

∑n=1∞2n(12n)α=∑n=1∞2n−nα=∑n=1∞2(1−α)n

(ii) is a geometric series with ratio 2(1−α). (ii) is finitely convergent if its ratio is less than one (namely α>1). Thus, (i) is finitely convergent if and only if α>1.

## Convergence of products

While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let {an}n=1∞ be a sequence of positive numbers. Then the infinite product ∏n=1∞(1+an) converges if and only if the series ∑n=1∞an converges. Also similarly, if 0<an<1 holds, then ∏n=1∞(1−an) approaches a non-zero limit if and only if the series ∑n=1∞an converges .

This can be proved by taking the logarithm of the product and using limit comparison test.[8]

## See also

* L'Hôpital's rule
* Shift rule

## References

1. ↑ Wachsmuth, Bert G.. "MathCS.org - Real Analysis: Ratio Test". http://www.mathcs.org/analysis/reals/numser/t_ratio.html.
2. ↑ František Ďuriš, Infinite series: Convergence tests, pp. 24–9. Bachelor's thesis.
3. ↑ Weisstein, Eric W.. "Bertrand's Test" (in en). https://mathworld.wolfram.com/BertrandsTest.html.
4. ↑ * Hazewinkel, Michiel, ed. (2001), "Gauss criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Gauss_criterion&oldid=39905
5. ↑ "Über die Convergenz und Divergenz der unendlichen Reihen.". Journal für die reine und angewandte Mathematik 1835 (13): 171–184. 1835-01-01. doi:10.1515/crll.1835.13.171. ISSN 0075-4102. https://www.degruyter.com/document/doi/10.1515/crll.1835.13.171/html.
6. ↑ Tong, Jingcheng (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. https://www.jstor.org/stable/2974907.
7. ↑ Samelson, Hans (1995). "More on Kummer's Test" (in en). The American Mathematical Monthly 102 (9): 817–818. doi:10.1080/00029890.1995.12004667. ISSN 0002-9890. https://www.tandfonline.com/doi/full/10.1080/00029890.1995.12004667.
8. ↑ Belk, Jim (26 January 2008). "Convergence of Infinite Products". https://cornellmath.wordpress.com/2008/01/26/convergence-of-infinite-products/.

## Further reading

* Leithold, Louis (1972). The Calculus, with Analytic Geometry (2nd ed.). New York: Harper & Row. pp. 655–737. ISBN 0-06-043959-9.

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