In
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
defines a
series that is denoted
:
The th
partial sum is the sum of the first terms of the sequence; that is,
:
A series is convergent (or converges) if the sequence
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
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
such that for every arbitrarily small positive number
, there is a (sufficiently large)
integer such that for all
,
:
If the series is convergent, the (necessarily unique) number
is called the ''sum of the series''.
The same notation
:
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):
*:
* Alternating the signs of the reciprocals of positive integers produces a convergent series (
alternating harmonic series):
*:
* 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):
*:
* The reciprocals of
triangular numbers produce a convergent series:
*:
* The reciprocals of
factorials produce a convergent series (see
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):
*:
* 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"):
*:
* The reciprocals of
powers of any n>1 produce a convergent series:
*:
* 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:
*:
* Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
*:
* The reciprocals of
Fibonacci numbers produce a convergent series (see
ψ):
*:
Convergence tests
There are a number of methods of determining whether a series converges or
diverges.
Comparison test. The terms of the sequence
are compared to those of another sequence
. If,
for all ''n'',
, and
converges, then so does
However,
if, for all ''n'',
, and
diverges, then so does
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'',
is not zero. Suppose that there exists
such that
:
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:
:
: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
be a positive and
monotonically decreasing function. If
:
then the series converges. But if the integral diverges, then the series does so as well.
Limit comparison test. If
, and the limit
exists and is not zero, then
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 ...
converges.
Alternating series test. Also known as the ''Leibniz criterion'', the
alternating series test states that for an
alternating series of the form
, if
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
is a positive monotone decreasing sequence, then
converges if and only if
converges.
Dirichlet's test
Abel's test
Conditional and absolute convergence
For any sequence
,
for all ''n''. Therefore,
:
This means that if
converges, then
also converges (but not vice versa).
If the series
converges, then the series
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
converges but the series
diverges, then the series
is
conditionally convergent. The Maclaurin series of the
logarithm function 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
be a sequence of functions.
The series
is said to converge uniformly to ''f''
if the sequence
of partial sums defined by
:
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
:
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
there is a positive integer
such that for all
we have
:
which is equivalent to
:
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)