Absolute convergence
In mathematics, a series is a sum of a sequence of terms.
Examples of simple series include arithmetic series which is a sum of a arithmetic progression which can be written as:
and geometric series which is a sum of a geometric progression which can be written as:
An infinite series is a sum of infinitely many terms. Such a sum can have a finite value; if it has, it is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes.
The simplest convergent infinite series is perhaps
It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2 -- in other words, the series has an upper bound.
This series is a geometric series and mathematicians usually write it as:
Formally, if an infinite series
is given with real (or complex) numbers an, we say that the series converges towards S or that its value is S if the limit
exists and is equal to S. If there is no such number, then the series is said to diverge.
History of the theory of infinite series
Convergence criteria
The investigation of the validity of infinite series is considered to begin
with Gauss. Euler had already considered the hypergeometric series
on which Gauss published a memoir in 1812.
celebrated memoir on the following series in 1812.It established the simpler criteria of convergence, and the questions of remainders and the range of
convergence.
Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long by Gregory (1668). Euler and Gauss had given various criteria, and Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.
Abel (1826) in his memoir on the series
corrected certain of Cauchy's conclusions, and gave a completely
scientific summation of the series for complex values of  and  . He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and
the same may be said of Raabe (1832), who made the first elaborate
investigation of the subject, of De Morgan (from 1842), whose
logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have
shown to fail within a certain region; of Bertrand (1842), Bonnet
(1843), Malmsten (1846, 1847, the latter without integration);
Stokes (1847), Paucker (1852), Tch\'ebichef (1852), and Arndt
(1853). General criteria began with Kummer (1835), and have been
studied by Eisenstein (1847), Weierstrass in his various
contributions to the theory of functions, Dini (1867),
DuBois-Reymond (1873), and many others. Pringsheim's (from 1889)
memoirs present the most complete general theory.
Uniform convergence
The theory of uniform convergence was treated by Cauchy (1821), his
limitations being pointed out by Abel, but the first to attack it
successfully were Stokes and Seidel (1847-48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomé used the
doctrine (1866), but there was great delay in recognizing the
importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.
Semi-convergence
Semi-convergent series were studied by Poisson (1823), who also gave
a general form for the remainder of the Maclaurin formula. The most
important solution of the problem is due, however, to Jacobi (1834),
who attacked the question of the remainder from a different
standpoint and reached a different formula. This expression was
also worked out, and another one given, by Malmsten (1847).
Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also
improved Jacobi's remainder, and showed the relation between the
remainder and Bernoulli's function  . Genocchi (1852) has further contributed to the theory.
Among the early writers was Wronski, whose "loi suprême" (1815)
was hardly recognized until Cayley (1873) brought it into
prominence.
Interpolation
Interpolation formulas have been given by various writers from
Newton to the present time. Lagrange's theorem is well known,
although Euler had already given an analogous form, as are also
Olivier's formula (1827), and those of Minding (1830), Cauchy
(1837), Jacobi (1845), Grunert (1850, 1853), Christoffel (1858), and
Mehler (1864).
Fourier series
Fourier series were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
Jakob Bernoulli (1702) and his brother Johann (1701) and still
earlier by Viète. Euler and Lagrange had simplified the subject,
as have, more recently, Poinsot, Schröter, Glaisher, and
Kummer. Fourier (1807) set for himself a different problem, to
expand a given function of in terms of the sines or cosines of
multiples of , a problem which he embodied in his Théorie
analytique de la Chaleur (1822). Euler had already given the
formulas for determining the coefficients in the series; and
Lagrange had passed over them without recognizing their value, but
Fourier was the first to assert and attempt to prove the general
theorem. Poisson (1820-23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for Cauchy (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner. Dirichlet's treatment (Crelle, 1829), while
bringing the theory of trigonometric series to a temporary
conclusion, has been the subject of criticism and improvement by
Riemann (1854), Heine, Lipschitz, Schläfli, and
DuBois-Reymond. Among other prominent contributors to the theory of
trigonometric and Fourier series have been Dini, Hermite, Halphen,
Krause, Byerly and Appell.
Some types of infinite series
- A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example: 1 + 1/2 + 1/4 + 1/8 + 1/16...
- The harmonic series is the series 1 + 1/2 + 1/3 + 1/4 + 1/5...
- An alternating series is a series where terms alternate signs. Example: 1 - 1/2 + 1/3 + 1/4 - 1/5...
Convergence criteria
- If the series ∑ an converges, then the sequence (an) converges to 0 for n→∞; the converse is in general not true.
- If all the numbers an are positive and ∑ bn is a convergent series such that an ≤ bn for all n, then ∑ an converges as well. If all the bn are positive, an ≥ bn for all n and ∑ bn diverges, then ∑ an diverges as well.
- If the an are positive and there exists a constant C < 1 such that an+1/an ≤ C, then ∑ an converges.
- If the an are positive and there exists a constant C < 1 such that (an)1/n ≤ C, then ∑ an converges.
- Integral test: if f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = an for all n, then ∑ an converges if and only if the integral ∫1∞ f(x) dx is finite.]
- A series of the form ∑ (-1)n an (with an ≥ 0) is called alternating. Such a series converges if the sequence an is monotone decreasing and converges towards 0. The converse is in general not true.
- See ratio test.
Examples
The series
converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion 5) from above.
As a function of r, the sum of this series is Riemann's zeta function.
The geometric series
converges if and only if |z| < 1.
The telescoping series
converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 - L.
Absolute convergence
The sum
-
is said to converge absolutely if the series of absolute values
converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.
If a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Even more: if the an are real and S is any real number, one can find a reordering so that the reordered series converges with limit S (Riemann).
Power series
Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. See also radius of convergence.
Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent.
When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
Generalizations
The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.
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