Convergence tests


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

List of tests

Limit of the summand">Term test">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.

[Root test]

This is also known as the nth root test or Cauchy's criterion.
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.
For example, for the series
convergence follows from the root test but not from the ratio test.

Integral test">Integral test for convergence">Integral test

The series can be compared to an integral to establish convergence or divergence. Let be a non-negative and monotonically decreasing function such that.

[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 finite and non-zero, then diverges if and only if diverges.
'''

[Cauchy condensation test]

Let be a positive non-increasing 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:
  1. is a convergent series,
  2. is a monotonic sequence, and
  3. is bounded.
Then is also convergent.

Absolute convergence test">Absolute convergence">Absolute convergence test

Every absolutely convergent series converges.

[Alternating series test]

This is also known as the Leibniz criterion.
Suppose the following statements are true:
  1. ,
  2. for every n,
Then and are convergent series.

[Dirichlet's test]

If is a sequence of real numbers and a sequence of complex numbers satisfying
where M is some constant, then the series
converges.

Raabe–Duhamel's test">Ratio test#2. Raabe's test">Raabe–Duhamel's test

Let.
Define
If
exists there are three possibilities:
An alternative formulation of this test is as follows. Let be a series of real numbers. Then if b > 1 and K exist such that
for all n > K then the series is convergent.

Bertrand's test">Ratio test#3. Bertrand’s test">Bertrand's test

Let be a sequence of positive numbers.
Define
If
exists, there are three possibilities:
Let be a sequence of positive numbers. If for some β > 1, then converges if and diverges if.

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. 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 non-zero limit if and only if the series converges.
This can be proved by taking the logarithm of the product and using limit comparison test.