Convergence tests
Part of a series of articles about  
Calculus  



Specialized


In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
Contents

1 List of tests
 1.1 Limit of the summand
 1.2 Ratio test
 1.3 Root test
 1.4 Integral 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 Raabe–Duhamel's test
 1.13 Bertrand's test
 1.14 Notes
 2 Comparison
 3 Examples
 4 Convergence of products
 5 See also
 6 References
 7 External links
List of tests
Limit of the summand
If the limit of the summand is undefined or nonzero, that is , 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.
Ratio test
This is also known as D'Alembert's criterion.
 Suppose that there exists such that
 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
 where denotes the limit superior (possibly ; if the limit exists it is the same value).
 If r < 1, then the series converges. 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 more powerful than the 'ratio test'
Integral test
The series can be compared to an integral to establish convergence or divergence. Let be a nonnegative and monotone decreasing function such that .
 If
 then the series converges. But if the integral diverges, then the series does so as well.
 In other words, the series converges if and only if the integral converges.
Direct comparison test
If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.
Limit comparison test
If , and the limit exists, is constant positive and finite and is not zero, then converges if and only if converges.
Cauchy condensation test
Let be a positive nonincreasing sequence. Then the sum converges if and only if the sum converges. Moreover, if they converge, then holds.
Abel's test
Suppose the following statements are true:
 is a convergent series,
 is a monotone sequence, and
 is bounded.
Then is also convergent.
Absolute convergence test
Every absolutely convergent series converges.
Alternating series test
Suppose the following statements are true:
 is a series whose terms alternate from positive to negative,
 ,
 the absolute value of each term is less than the absolute value of the previous term.
Then is convergent.
Remark: This is also known as the Leibniz criterion.
Dirichlet's test
If is a sequence of real numbers and a sequence of complex numbers satisfying

 for every positive integer N
where M is some constant, then the series
converges.
Raabe–Duhamel's test
Let .
Define
If
exists there are three possibilities:
 if L > 1 the series converges
 if L < 1 the series diverges
 and if L = 1 the test is inconclusive.
An alternative formulation of this test is as follows. Let { a_{n} } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
for all n > K then the series {a_{n}} is convergent.
Bertrand's test
Let { a_{n} } > 0.
Define
If
exists there are three possibilities:
 if L > 1 the series converges
 if L < 1 the series diverges
 and if L = 1 the test is inconclusive.
More information about Bertrand's test (Page 24).
Notes
 For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.
Comparison
The root test is stronger than the ratio test (it is more powerful because the required condition is weaker): whenever the ratio test determines the convergence or divergence /of an infinite series, the root test does too, but not conversely.^{[1]}
For example, for the series
 1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 + 0.125 + 0.125 + ... = 4
convergence follows from the root test but not from the ratio test.^{[2]}
Examples
Consider the series
.
Cauchy condensation test implies that (*) is finitely convergent if
is finitely convergent. Since
(**) is geometric series with ratio . (**) is finitely convergent if its ratio is less than one (namely ). Thus, (*) is finitely convergent if and only if .
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 be a sequence of positive numbers. Then the infinite product converges if and only if the series converges. Also similarly, if holds, then approaches a nonzero limit if and only if the series converges .
This can be proved by taking the logarithm of the product and using limit comparison test.^{[3]}
See also
References
External links
 Flowchart for choosing convergence test