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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

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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 d'Alembert's criterion.

Suppose that there exists   such that
 
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge.
The power series 1, .5, .25, .125,... has   and hence converges. If the series were   it would diverge. If it were   the ratio test would fail to give an answer (though obviously that series does diverge).

This is also known as the nth root test or Cauchy's criterion.

Let
 
where   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.

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]

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.[2]

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. Thus, the series   converges if and only if the integral converges.
The harmonic series 1, 1/2, 1/3, 1/4,... has   and so does not converge. The integral test succeeds here where the ratio test fails, because  . The root test also fails, because  

If the series   is an absolutely convergent series and   for sufficiently large n , then the series   converges absolutely.

The harmonic series   1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8,... can be compared to the series   1/2, 1/4, 1/4, 1/8, 1/8, 1/8, 1/8, 1/16,..., where   for any   The sum of the   is infinite because it is 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8)+ (1/16 +..., an infinite sum of 1/2's, so the sum of the bigger   must also be infinite. As noted above, the ratio test and root test fail for the harmonic series.

If  , (that is, each element of the two sequences is positive) and the limit   exists, is finite and non-zero, then   diverges if and only if   diverges.

Let   be a positive non-increasing sequence. Then the sum   converges if and only if the sum   converges. Moreover, if they converge, then   holds.

Suppose the following statements are true:

  1.   is a convergent series,
  2.   is a monotonic sequence, and
  3.   is bounded.

Then   is also convergent.

Every absolutely convergent series converges.

This is also known as the Leibniz criterion.

Suppose the following statements are true:

  1.  ,
  2. for every n,  

Then   and   are convergent series.

If   is a sequence of real numbers and   a sequence of complex numbers satisfying

  •  
  •  
  •   for every positive integer N

where M is some constant, then the series

 

converges.

Let  .

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 { 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:[3][4]

  • if L > 1 the series converges
  • if L < 1 the series diverges
  • and if L = 1 the test is inconclusive.

Let { an } be a sequence of positive numbers. If   for some β > 1, then   converges if α > 1 and diverges if α ≤ 1.[5]

Notes

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  • For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Examples

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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  .

  1. ^ Wachsmuth, Bert G. "MathCS.org - Real Analysis: Ratio Test". www.mathcs.org.
  2. ^ In the example of S = 1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 + 0.125 + 0.125 + ..., the ratio test is inconclusive if   is odd so   (though not if   is even), because it looks at
     
    The root test shows convergence because it looks at
     
  3. ^ František Ďuriš, Infinite series: Convergence tests, pp. 24–9. Bachelor's thesis.
  4. ^ Weisstein, Eric W. "Bertrand's Test". mathworld.wolfram.com. Retrieved 2020-04-16.
  5. ^ * "Gauss criterion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]