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. 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.

[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.

[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. 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.

-series test

A commonly used corollary of the integral test is the p-series test. Let. Then converges if.
The case of yields the harmonic series, which diverges. The case of is the Basel problem and the series converges to. In general, for, the series is equal to the Riemann zeta function applied to, that is.

[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 either both series converge or both series diverge.

[Cauchy condensation test]

Let be a non-negative 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]

Suppose the following statements are true:
  • is monotonic,
Then and are convergent series.
This test is also known as the Leibniz criterion.

[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.

[Cauchy's convergence test]

A series is convergent if and only if for every there is a natural number N such that
holds for all and all.

[Stolz–Cesàro theorem]

Let and be two sequences of real numbers. Assume that is a strictly monotone and divergent sequence and the following limit exists:
Then, the limit

[Weierstrass M-test]

Suppose that is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers satisfying the conditions
  • for all and all, and
  • converges.
Then the series
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">Ratio test#2. Raabe's test">Raabe–Duhamel's test

Let be a sequence of positive numbers.
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 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:
  • if L > 1 the series converges
  • if L < 1 the series diverges
  • and if L = 1 the test is inconclusive.

Gauss's test">Ratio test#5. Gauss's test">Gauss's test

Let be a sequence of positive numbers. If for some β > 1, then converges if and diverges if.

Kummer's test">Ratio test#6. Kummer's test">Kummer's test

Let be a sequence of positive numbers. Then:
converges if and only if there is a sequence of positive numbers and a real number c > 0 such that.
diverges if and only if there is a sequence of positive numbers such that
and diverges.

Abu-Mostafa's test

Let be an infinite series with real terms and let be any real function such that for all positive integers n and the second derivative exists at. Then converges absolutely if and diverges otherwise.

Examples

Consider the series
Cauchy condensation test implies that is finitely convergent if
is finitely convergent. Since
is a geometric series with ratio. is finitely convergent if its ratio is less than one 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.