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 1

Limit of the summand 1.1

Ratio test 1.2

Root test 1.3

Integral test 1.4

Direct comparison test 1.5

Limit comparison test 1.6

Cauchy condensation test 1.7

Abel's test 1.8

Alternating series test 1.9

Dirichlet's test 1.10

RaabeDuhamel's test 1.11

Notes 1.12

Comparison 2

Examples 3

Convergence of products 4

See also 5

References 6

External links 7
List of tests
If the limit of the summand is undefined or nonzero, that is \lim_{n \to \infty}a_n \ne 0, 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 r such that

\lim_{n \to \infty} \left\frac{a_{n+1}}{a_n}\right = r.

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.
This is also known as the nth root test or Cauchy's criterion. Define r as follows:

r = \limsup_{n \to \infty}\sqrt[n]{a_n},

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.
The series can be compared to an integral to establish convergence or divergence. Let f:[1,\infty)\to\R_+ be a positive and monotone decreasing function such that f(n) = a_n. If

\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,

then the series converges. But if the integral diverges, then the series does so as well.

In other words, the series {a_n} converges if and only if the integral converges.
If the series \sum_{n=1}^\infty b_n is an absolutely convergent series and a_n\le b_n for sufficiently large n , then the series \sum_{n=1}^\infty a_n converges absolutely.
If \left \{ a_n \right \}, \left \{ b_n \right \} > 0, and the limit \lim_{n \to \infty} \frac{a_n}{b_n} exists, is finite and is not zero, then \sum_{n=1}^\infty a_n converges if and only if \sum_{n=1}^\infty b_n converges.
Let \left \{ a_n \right \} be a positive nonincreasing sequence. Then the sum A = \sum_{n=1}^\infty a_n converges if and only if the sum A^* = \sum_{n=0}^\infty 2^n a_{2^n} converges. Moreover, if they converge, then A \leq A^* \leq 2A holds.
Suppose the following statements are true:

\sum a_n is a convergent series,

{b_{n}} is a monotone sequence, and

{b_{n}} is bounded.
Then \sum a_nb_n is also convergent.
This is also known as the Leibniz criterion. If \sum_{n=1}^\infty a_n is a series whose terms alternative from positive to negative, and if the limit as n approaches infinity of a_n is zero and the absolute value of each term is less than the absolute value of the previous term, then \sum_{n=1}^\infty a_n is convergent.
Let { a_{n} } > 0.
Define
b_n = n \left( \frac{ a_n }{ a_{ n + 1 } }  1 \right ) .
If
L = \lim_{ n \to \infty } b_n
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
\frac{ a_{ n + 1 } }{ a_n } \le 1  \frac{ b }{ n }
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
(*) \;\;\; \sum_{n=1}^{\infty} \frac{1}{n^\alpha}.
Cauchy condensation test implies that (*) is finitely convergent if
(**) \;\;\; \sum_{n=1}^{\infty} 2^n \left ( \frac{1}{2^n}\right )^\alpha
is finitely convergent. Since
\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}
(**) is geometric series with ratio 2^{(1\alpha)} . (**) is finitely convergent if its ratio is less than one (namely \alpha > 1). Thus, (*) is finitely convergent if and only if \alpha > 1 .
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 \left \{ a_n \right \}_{n=1}^\infty be a sequence of positive numbers. Then the infinite product \prod_{n=1}^\infty (1 + a_n) converges if and only if the series \sum_{n=1}^\infty a_n converges. Also similarly, if 0 < a_n < 1 holds, then \prod_{n=1}^\infty (1  a_n) approaches a nonzero limit if and only if the series \sum_{n=1}^\infty a_n converges .
This can be proved by taking logarithm of the product and using limit comparison test.^{[2]}
See also
References

^ Ratio Test

^ Convergence of Infinite Products
External links

Flowchart for choosing convergence test
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