Power Series

Definition: Power Series

Power Series

Noun

1. The sum of terms containing successively higher integral powers of a variable.

Specialty Definition: Power Series

Domain	Definition
Aerospace	An infinite series of increasing power of the variable, of the form anxn = a0 + a1x + a2x2 . . . + anxnBoth the variable and the coefficients may take on complex values. The totality of values of x for which a power series is convergent is called the interval of convergence of the series. (references)
Source: compiled by the editor from various references; see credits.

Specialty Definition: Power series

In mathematics, a power series is an infinite series of the form

where the coefficients a_n, the center a, and the argument x are real or complex numbers. These series usually arise as the Taylor series of some known function; the Taylor series article contains many examples.

Radius of convergence

A power series will converge for some values of the variable x (at least for x = a) and may diverge for others. It turns out that there is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x - a| < r and diverges whenever |x - a| > r. (For |x - a| = r we cannot make any general statement.) The number r is called the radius of convergence of the power series; in general it is given as

r = lim inf_{n → ∞} |a_n|^-1/n

but a fast way to compute it is

r = lim_{n → ∞} |a_n/a_n+1|.

The latter formula is valid only if the limit exists, while the former formula can always be used.

The series converges absolutely for |x - a| < r and converges uniformly on every compact subset of {x : |x - a| < r}.

Differentiating and integrating power series

Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set. It can be differentiated and integrated quite easily, by treating every term separately:

Both of these series have the same radius of convergence as the original one.

Analytic functions

A function f defined on some open subset U of R or C is called analytic if it is locally given by power series. This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a which converges to f(x) for every x ∈ V.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.

If a function is analytic, then it is infinitely often differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients a_n can be computed as

where f⁽ⁿ⁾(a) denotes the n-th derivative of f at a. This means that every analytic function is locally represented by its Taylor series.

The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element a∈U such that f⁽ⁿ⁾(a) = g⁽ⁿ⁾(a) for all n ≥ 0, then f(x) = g(x) for all x ∈ U.

If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than { x : |x - a| < r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with |x - a| = r such that no analytic continuation of the series can be defined at x.

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

Formal power series

In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a principle that is of great utility in combinatorics.

Note that the "is an element of" symbol, appears as a square on some fonts (such as the default display font of windows)

Modern Usage: Power Series

Domain	Usage
Screenplays	Now, if you take the sum of the integrants and express the result as a power series, then the indices show the basic binary blocks, only I wouldn't do it if I were you! (Doctor Who; writing credit: Basil Caplan; Martin Defalco)
Source: compiled by the editor from various references; see credits.

Commercial Usage: Power Series

Domain	Title
Books	Catherine De'Medici (Profiles in Power Series (Paper)) (reference) Catherine the Great: Profiles in Power Series (reference) Constituting Identity: Political Identity Formation and the Constitution in Post-Independence Ireland (Law, Justice, and Power Series) (reference) Delta: America's Elite Counterterrorist Force (The Power Series) (reference) The Great Elector: Frederick William of Brandenburg - Prussia (Profiles in Power Series) (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Frequency of Internet Keywords: Power Series

Modern Translation: Power Series

Language		Translations for "power series"; alternative meanings/domain in parentheses.
Hungarian		hatványsor. (various references)

Pig Latin		owerpay eriessay

Swedish		potensserie. (various references)
Source: compiled by the editor from various translation references.

Anagrams: Power Series

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "e-e-e-i-o-p-r-r-s-s-w"
	-3 letters: prioress, priseres, reposers, repowers, reprises, respires, roperies, roseries, sweepers, sweepier.
	-4 letters: orrises, peeress, peeries, perries, poesies, poisers, presser, pressor, prisere, prosers, prosier, prossie, prowess, reposer, reposes, repower, repress, reprise, rerises, respire, rewires, seepier, serries, sirrees, soirees, spewers, sweeper, weepers, weepier, weepies, worries.
	-5 letters: eerier, eposes, eroses, espies, osiers, peerie, peises, perses, pewees, pisser.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: Power Series

Hexadecimal (or equivalents, 770AD-1900s) (references)

50 6F 77 65 72 53 65 72 69 65 73

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01010000 01101111 01110111 01100101 01110010 00100000 01010011 01100101 01110010 01101001 01100101 01110011

HTML Code (1990) (references)

&#80 &#111 &#119 &#101 &#114 &#32 &#83 &#101 &#114 &#105 &#101 &#115

ISO 10646 (1991-1993) (references)

0050 006F 0077 0065 0072 0053 0065 0072 0069 0065 0073

Encryption (beginner's substitution cypher): (references)

Expression	Frequency per Day
power series	14
400 antec atx power series sl400 solution supply watt	4
gt power series	4
austin power series	3
3.0 gt power series	3
power series stone	2
8bh high power series speed	2
Source: compiled by the editor from various references; see credits.

Power Series

Definition: Power Series

Power Series

Noun

Specialty Definition: Power Series

Aerospace

Specialty Definition: Power series

Radius of convergence

Differentiating and integrating power series

Analytic functions

Formal power series

Crosswords: Power Series

Modern Usage: Power Series

Screenplays

Commercial Usage: Power Series

Books

Frequency of Internet Keywords: Power Series

power series

400 antec atx power series sl400 solution supply watt

gt power series

austin power series

3.0 gt power series

power series stone

8bh high power series speed

Modern Translation: Power Series

Hungarian

Pig Latin

Swedish

Anagrams: Power Series

Scrabble® Enable2K-Verified Anagrams

Alternative Orthography: Power Series

50 6F 77 65 72 53 65 72 69 65 73

01010000 01101111 01110111 01100101 01110010 00100000 01010011 01100101 01110010 01101001 01100101 01110011

&#80 &#111 &#119 &#101 &#114 &#32 &#83 &#101 &#114 &#105 &#101 &#115

0050 006F 0077 0065 0072 0053 0065 0072 0069 0065 0073

50818971842537184757185

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