In
mathematics, the antilimit is the equivalent of 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 ...
for a
divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mu ...
. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
.
Common divergent series
See also
*
Abel summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must ...
*
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean
) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as ''n'' tends to infinity, of ...
*
Lindelöf summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series mus ...
*
Euler summation
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ'' ...
*
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several var ...
*
Mittag-Leffler summation In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by
Definition
Let
:y(z) = \sum_^\infty y_kz^k
be a formal power series in ''z''.
...
*
Lambert summation In mathematical analysis, Lambert summation is a summability method for a class of divergent series.
Definition
A series \sum a_n is ''Lambert summable'' to ''A'', written \sum a_n = A \,(\mathrm), if
:\lim_ (1-r) \sum_^\infty \frac = A .
If a s ...
*
Euler–Boole summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by
: \frac=\sum_^\infty E_n(x)\frac.
The concept is named after Leonhard Euler and George Boole
George Boole (; 2 November ...
and
Van Wijngaarden transformation In mathematics and numerical analysis, the van Wijngaarden transformation is a variant on the Euler transform used to accelerate the convergence of an alternating series
In mathematics, an alternating series is an infinite series of the form ...
can also be used on divergent series
Reference
*
*
{{Series (mathematics)
Divergent series
Summability methods
Sequences and series
Mathematical analysis