Part of a series of articles about 
Calculus 

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
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 nthterm test, test for divergence, or the divergence test.
This is also known as d'Alembert's criterion.
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.^{[1]}
The series can be compared to an integral to establish convergence or divergence. Let be a nonnegative and monotonically decreasing function such that . If
A commonlyused corollary of the integral test is the pseries 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 .
If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.
If , (that is, each element of the two sequences is positive) and the limit exists, is finite and nonzero, then either both series converge or both series diverge.
Let be a nonnegative nonincreasing sequence. Then the sum converges if and only if the sum converges. Moreover, if they converge, then holds.
Suppose the following statements are true:
Then is also convergent.
Every absolutely convergent series converges.
Suppose the following statements are true:
Then and are convergent series. This test is also known as the Leibniz criterion.
If is a sequence of real numbers and a sequence of complex numbers satisfying
where M is some constant, then the series
converges.
A series is convergent if and only if for every there is a natural number N such that
holds for all n > N and all p ≥ 1.
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
Suppose that (f_{n}) is a sequence of real or complexvalued functions defined on a set A, and that there is a sequence of nonnegative numbers (M_{n}) satisfying the conditions
Then the series
converges absolutely and uniformly on A.
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.
Let { a_{n} } be a sequence of positive numbers.
Define
If
exists there are three possibilities:
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.
Let { a_{n} } be a sequence of positive numbers.
Define
If
exists, there are three possibilities:^{[2]}^{[3]}
Let { a_{n} } be a sequence of positive numbers. If for some β > 1, then converges if α > 1 and diverges if α ≤ 1.^{[4]}
Let { a_{n} } be a sequence of positive numbers. Then:^{[5]}^{[6]}^{[7]}
(1) converges if and only if there is a sequence of positive numbers and a real number c > 0 such that .
(2) diverges if and only if there is a sequence of positive numbers such that
and diverges.
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.^{[8]}
Consider the series

(i)

Cauchy condensation test implies that (i) is finitely convergent if

(ii)

is finitely convergent. Since
(ii) is a geometric series with ratio . (ii) is finitely convergent if its ratio is less than one (namely ). Thus, (i) is finitely convergent if and only if .
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.^{[9]}