In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence

Absolute convergence

In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

, interval of convergence or divergence of an infinite series

Series (mathematics)

A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

.

List of tests

• 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
  Cauchy sequence
  In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
  
   only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero.

• Ratio test (d'Alembert's criterion). Suppose that there exists such that

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

• Root test or nth root test. Define r as follows:

where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

• Integral test
  Integral test for convergence
  In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. An early form of the test of convergence was developed in India by Madhava in the 14th century, and by his followers at the Kerala School...
  
  . The series can be compared to an integral to establish convergence or divergence. Let be a positive and monotone decreasing function
  Monotonic function
  In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
  
   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

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 the integral converges.

• Direct comparison test
  Comparison test
  In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
  
  . If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.

• Limit comparison test
  Limit comparison test
  In mathematics, the limit comparison test is a method of testing for the convergence of an infinite series.- Statement :...
  
  . If , and the limit exists and is not zero, then converges if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
    converges.

• Cauchy condensation test. Let be a positive non-increasing sequence. Then the sum converges if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   the sum converges. Moreover, if they converge, then holds.

• Abel's test
  Abel's test
  In mathematics, Abel's test is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Abel...
  
  

• Alternating series test
  Alternating series test
  The alternating series test is a method used to prove that infinite series of terms converge. It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or the Leibniz criterion.A series of the form...
  
   (Leibniz criterion)

• For some specific types of series there are more specialized convergence tests, for instance for Fourier series
  Fourier series
  In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
  
   there is the Dini test
  Dini test
  In mathematics, the Dini and Dini-Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.- Definition :...
  
  .

Comparison

The root test is stronger than the ratio test (it is more powerful because the required condition is weaker): 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

1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 + 0.125 + 0.125 + ...=4

convergence follows from the root test but not from the ratio test.

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 (namely ). Thus, (*) is finitely convergent if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 .

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

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

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 logarithm of the product and using limit comparison test.

External links

• Flowchart for choosing convergence test
• Convergence of infinite series