If the limit of the summand is undefined or nonzero, that is
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 root test or Cauchy's criterion.
[1] The series can be compared to an integral to establish convergence or divergence.
be a non-negative and monotonically decreasing function such that
But if the integral diverges, then the series does so as well.
yields the harmonic series, which diverges.
is the Basel problem and the series converges to
, the series is equal to the Riemann zeta function applied to
for sufficiently large n , then the series
, (that is, each element of the two sequences is positive) and the limit
a sequence of complex numbers satisfying where M is some constant, then the series converges.
is a strictly monotone and divergent sequence and the following limit exists: Then, the limit 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 Then the series converges absolutely and uniformly on A.
Extensions to the ratio test, however, sometimes allows one to deal with this case.
Let { an } be a sequence of positive numbers.
Define If exists there are three possibilities: 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 for all n > K then the series {an} is convergent.
Let { an } be a sequence of positive numbers.
Define If exists, there are three possibilities:[2][3] Let { an } be a sequence of positive numbers.
[4] Let { an } be a sequence of positive numbers.
be an infinite series with real terms and let
[8] Consider the series Cauchy condensation test implies that (i) is finitely convergent if is finitely convergent.
Since (ii) is a geometric series with ratio
(ii) is finitely convergent if its ratio is less than one (namely
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.
be a sequence of positive numbers.
approaches a non-zero limit if and only if the series
This can be proved by taking the logarithm of the product and using limit comparison test.