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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = \sum_^n a_k. A series is convergent (or converges) if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , < \varepsilon. If the series is convergent, the (necessarily unique) number \ell is called the ''sum of the series''. The same notation :\sum_^\infty a_k is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: denotes the ''operation of adding and '' as well as the result of this ''addition'', which is called the ''sum'' of and . Any series that is not convergent is said to be divergent or to diverge.


Examples of convergent and divergent series

* The reciprocals of the
positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
produce a divergent series ( harmonic series): *: ++++++\cdots \rightarrow \infty. * Alternating the signs of the reciprocals of positive integers produces a convergent series ( alternating harmonic series): *:-+-+-\cdots = \ln(2) * The reciprocals of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s produce a divergent series (so the set of primes is "
large Large means of great size. Large may also refer to: Mathematics * Arbitrarily large, a phrase in mathematics * Large cardinal, a property of certain transfinite numbers * Large category, a category with a proper class of objects and morphisms ...
"; see divergence of the sum of the reciprocals of the primes): *: ++++++\cdots \rightarrow \infty. * The reciprocals of triangular numbers produce a convergent series: *: ++++++\cdots = 2. * The reciprocals of factorials produce a convergent series (see e): *: \frac + \frac + \frac + \frac + \frac + \frac + \cdots = e. * The reciprocals of
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
s produce a convergent series (the Basel problem): *: ++++++\cdots = . * The reciprocals of
powers of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negati ...
produce a convergent series (so the set of powers of 2 is " small"): *: ++++++\cdots = 2. * The reciprocals of powers of any n>1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of
powers of 2 A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negati ...
also produces a convergent series: *: -+-+-+\cdots = . * Alternating the signs of reciprocals of powers of any n>1 produces a convergent series: *: -+-+-+\cdots = . * The reciprocals of Fibonacci numbers produce a convergent series (see ψ): *: \frac + \frac + \frac + \frac + \frac + \frac + \cdots = \psi.


Convergence tests

There are a number of methods of determining whether a series converges or diverges. Comparison test. The terms of the sequence \left \ are compared to those of another sequence \left \. If, for all ''n'', 0 \le \ a_n \le \ b_n, and \sum_^\infty b_n converges, then so does \sum_^\infty a_n. However, if, for all ''n'', 0 \le \ b_n \le \ a_n, and \sum_^\infty b_n diverges, then so does \sum_^\infty a_n.
Ratio test In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...
. Assume that for all ''n'', a_n is not zero. Suppose that there exists r such that :\lim_ \left, \ = r. If ''r'' < 1, then the series is absolutely convergent. If then the series diverges. If the ratio test is inconclusive, and the series may converge or diverge. Root test or ''n''th root test. Suppose that the terms of the sequence in question are non-negative. Define ''r'' as follows: :r = \limsup_\sqrt :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 then the series diverges. If the root test is inconclusive, and the series may converge or diverge. The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series. Integral test. The series can be compared to an integral to establish convergence or divergence. Let f(n) = a_n be a positive and monotonically decreasing function. If :\int_^ f(x)\, dx = \lim_ \int_^ f(x)\, dx < \infty, then the series converges. But if the integral diverges, then the series does so as well. Limit comparison test. If \left \, \left \ > 0, and the limit \lim_ \frac exists and is not zero, then \sum_^\infty a_n converges
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
\sum_^\infty b_n converges. Alternating series test. Also known as the ''Leibniz criterion'', the alternating series test states that for an alternating series of the form \sum_^\infty a_n (-1)^n, if \left \ is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.
Cauchy condensation test In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f(n) of non-negative real numbers, the series \sum\limits_^ f(n) converges if an ...
. If \left \ is a positive monotone decreasing sequence, then \sum_^\infty a_n converges if and only if \sum_^\infty 2^k a_ converges. Dirichlet's test Abel's test


Conditional and absolute convergence

For any sequence \left \, a_n \le \left, a_n \ for all ''n''. Therefore, :\sum_^\infty a_n \le \sum_^\infty \left, a_n \. This means that if \sum_^\infty \left, a_n \ converges, then \sum_^\infty a_n also converges (but not vice versa). If the series \sum_^\infty \left, a_n \ converges, then the series \sum_^\infty a_n is absolutely convergent. The
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
is absolutely convergent for every complex value of the variable. If the series \sum_^\infty a_n converges but the series \sum_^\infty \left, a_n \ diverges, then the series \sum_^\infty a_n is conditionally convergent. The Maclaurin series of the logarithm function \ln(1+x) is conditionally convergent for . The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.


Uniform convergence

Let \left \ be a sequence of functions. The series \sum_^\infty f_n is said to converge uniformly to ''f'' if the sequence \ of partial sums defined by : s_n(x) = \sum_^n f_k (x) converges uniformly to ''f''. There is an analogue of the comparison test for infinite series of functions called the Weierstrass M-test.


Cauchy convergence criterion

The Cauchy convergence criterion states that a series :\sum_^\infty a_n converges
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
the sequence of partial sums is a Cauchy sequence. This means that for every \varepsilon > 0, there is a positive integer N such that for all n \geq m \geq N we have : \left, \sum_^n a_k \ < \varepsilon, which is equivalent to :\lim_ \sum_^ a_k = 0.


See also

* Normal convergence * List of mathematical series


External links

* * Weisstein, Eric (2005)
Riemann Series Theorem
Retrieved May 16, 2005. {{Series (mathematics) Mathematical series Convergence (mathematics)