Wikipedia / Convergence tests
Wikipedia / 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
Term testLimit of the summand
, 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 testRatio test
This is also known as d'Alembert's criterion.
then the test is inconclusive, and the series may converge absolutely or conditionally or diverge.
Root testRoot test
This is also known as the nth root test or Cauchy's criterion.
Let
denotes the limit superior; 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.
Integral testIntegral test
be a non-negative and monotonically decreasing function converges if and only if the integral converges.
p-series test
.
is the Basel problem, the series is equal to the Riemann zeta function.
Direct comparison testDirect comparison test
is an absolutely convergent for sufficiently large n converges absolutely.
Limit comparison testLimit comparison test
exists, is finite and non-zero, then either both series converge or both series diverge.
Cauchy condensation testCauchy condensation test
converges if and only if holds.
Abel's testAbel's test
Suppose the following statements are true:
is a convergent series,
is a monotonic sequence, and
is bounded.
is also convergent.
Absolute convergenceAbsolute convergence test
Every absolutely convergent series converges.
Alternating series testAlternating series test
Suppose the following statements are true:
is monotonic,
are convergent series. This test is also known as the Leibniz criterion.
Dirichlet's testDirichlet's test
is a sequence of real numbers a sequence of complex numbers satisfying
for every positive integer N
where M is some constant, then the series
converges.
Cauchy's convergence testCauchy's convergence test
there is a natural number N such that
holds for all and all .
Stolz–Cesàro theoremStolz–Cesàro theorem
is a strictly monotone and divergent sequence and the following limit exists:
Then, the limit
Weierstrass M-testWeierstrass 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
, 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.
Ratio test#2. Raabe's testRaabe–Duhamel's test
Let { an } be a sequence of positive numbers.
Define
If
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 } 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 {an} is convergent.
Ratio test#3. Bertrand's testBertrand's test
Let { an } be a sequence of positive numbers.
Define
If
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.
Ratio test#5. Gauss's testGauss's test
Let { an converges if and diverges if .
Ratio test#6. Kummer's testKummer's test
Let { an } be a sequence of positive numbers. Then:
of positive numbers and a real number c.
diverges.
Abu-Mostafa's test
for all positive integers n and diverges otherwise.
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
Cauchy condensation test implies that () is finitely convergent if
is finitely convergent. Since
(. () 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 converges if and only if converges .
This can be proved by taking the logarithm of the product and using limit comparison test.