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Convergence tests
Calculus  



Specialized 

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series.
Contents
List of tests
Limit of the summand
If the limit of the summand is undefined or nonzero, that is <math>\lim_{n \to \infty}a_n \ne 0</math>, 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.
Ratio test
This is also known as D'Alembert's criterion. Suppose that there exists <math>r</math> such that
 <math>\lim_{n \to \infty} \left\frac{a_{n+1}}{a_n}\right = r.</math>
 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 or diverge.
Root test
This is also known as the nth root test or Cauchy's criterion. Define r as follows:
 <math>r = \limsup_{n \to \infty}\sqrt[n]{a_n},</math>
 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
The series can be compared to an integral to establish convergence or divergence. Let <math>f:[1,\infty)\to\R_+</math> be a nonnegative and monotone decreasing function such that <math>f(n) = a_n</math>. If
 <math>\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,</math>
 then the series converges. But if the integral diverges, then the series does so as well.
 In other words, the series <math>{a_n}</math> converges if and only if the integral converges.
Direct comparison test
If the series <math>\sum_{n=1}^\infty b_n</math> is an absolutely convergent series and <math>a_n\le b_n</math> for sufficiently large n , then the series <math>\sum_{n=1}^\infty a_n</math> converges absolutely.
Limit comparison test
If <math>\left \{ a_n \right \}, \left \{ b_n \right \} > 0</math>, and the limit <math>\lim_{n \to \infty} \frac{a_n}{b_n}</math> exists, is finite and is not zero, then <math>\sum_{n=1}^\infty a_n</math> converges if and only if <math>\sum_{n=1}^\infty b_n</math> converges.
Cauchy condensation test
Let <math>\left \{ a_n \right \}</math> be a positive nonincreasing sequence. Then the sum <math>A = \sum_{n=1}^\infty a_n</math> converges if and only if the sum <math>A^* = \sum_{n=0}^\infty 2^n a_{2^n}</math> converges. Moreover, if they converge, then <math>A \leq A^* \leq 2A</math> holds.
Abel's test
Suppose the following statements are true:
 <math>\sum a_n </math> is a convergent series,
 {b_{n}} is a monotone sequence, and
 {b_{n}} is bounded.
Then <math>\sum a_nb_n </math> is also convergent.
Alternating series test
This is also known as the Leibniz criterion. If <math>\sum_{n=1}^\infty a_n</math> is a series whose terms alternative from positive to negative, and if the limit as n approaches infinity of <math> a_n </math> is zero and the absolute value of each term is less than the absolute value of the previous term, then <math>\sum_{n=1}^\infty a_n</math> is convergent.
Dirichlet's test
RaabeDuhamel's test
Let { a_{n} } > 0.
Define
<math> b_n = n \left( \frac{ a_n }{ a_{ n + 1 } }  1 \right ) </math>.
If
<math> L = \lim_{ n \to \infty } b_n </math>
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 { a_{n} } be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
<math> \frac{ a_{ n + 1 } }{ a_n } \le 1  \frac{ b }{ n } </math>
for all n > K then the series { a_{n} } is convergent.
Notes
 For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test
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.^{[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.
Examples
Consider the series
<math>(*) \;\;\; \sum_{n=1}^{\infty} \frac{1}{n^\alpha}</math>.
Cauchy condensation test implies that (*) is finitely convergent if
<math> (**) \;\;\; \sum_{n=1}^{\infty} 2^n \left ( \frac{1}{2^n}\right )^\alpha </math>
is finitely convergent. Since
<math>\sum_{n=1}^{\infty} 2^n \left ( \frac{1}{2^n}\right )^\alpha = \sum_{n=1}^{\infty} 2^{nn\alpha} = \sum_{n=1}^{\infty} 2^{(1\alpha) n} </math>
(**) is geometric series with ratio <math> 2^{(1\alpha)} </math>. (**) is finitely convergent if its ratio is less than one (namely <math>\alpha > 1</math>). Thus, (*) is finitely convergent if and only if <math> \alpha > 1 </math>.
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 <math>\left \{ a_n \right \}_{n=1}^\infty</math> be a sequence of positive numbers. Then the infinite product <math>\prod_{n=1}^\infty (1 + a_n)</math> converges if and only if the series <math>\sum_{n=1}^\infty a_n</math> converges. Also similarly, if <math>0 < a_n < 1</math> holds, then <math>\prod_{n=1}^\infty (1  a_n)</math> approaches a nonzero limit if and only if the series <math>\sum_{n=1}^\infty a_n</math> converges .
This can be proved by taking logarithm of the product and using limit comparison test.^{[2]}