convergence (mathematics)
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; 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 and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large)
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
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"; 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
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
s produce a convergent series (see e): *: \frac + \frac + \frac + \frac + \frac + \frac + \cdots = e. * The reciprocals of square numbers produce a convergent series (the Basel problem): *: ++++++\cdots = . * The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "
small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...
"): *: ++++++\cdots = 2. * The reciprocals of powers of any n>1 produce a convergent series: *: ++++++\cdots = . * Alternating the signs of reciprocals of powers of 2 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 number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
s 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. 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
\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. 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

If the series \sum_^\infty \left, a_n \ converges, then the series \sum_^\infty a_n is said to be absolutely convergent. Every absolute convergent series (real or complex) is also convergent, but the converse is not true. The Maclaurin series of the exponential function is absolutely convergent for every
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 (see the Mercator series). 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. Agnew's theorem characterizes rearrangements that preserve convergence for all series.


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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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. This is equivalent to \lim_ \left(\sup_ \left, \sum_^ a_k \ \right) = 0.


See also

* Normal convergence *
List of mathematical series This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. *Here, 0^0 Zero to the power of zero, is taken to have the value 1 *\ denotes the fractional part ...


External links

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