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:$f(x)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; a\_n\; \backslash left(\; x-c\; ight)^n\; =\; a\_0\; +\; a\_1\; (x-c)\; +\; a\_2\; (x-c)^2\; +\; a\_3\; (x-c)^3\; +\; \backslash cdots$ where ''an'' represents the coefficient of the nth term, ''c'' is a constant, and ''x'' varies around ''c'' (for this reason one sometimes speaks of the series as being ''centered'' at ''c''). The numbers ''a'', ''c'', and ''x'' are usually Real or Complex . This series usually arises as the Taylor Series of some known Function ; the Taylor Series article contains many examples. In many situations ''c'' is equal to zero, for instance when considering a Maclaurin Series . In such cases, the power series takes the simpler form ::$$ f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots. These power series arise primarily in analysis, but also occur in combinatorics (under the name of Generating Function s) and in electrical engineering (under the name of the Z-transform ). The familiar Decimal Notation for Integer s can also be viewed as an example of a power series, but with the argument ''x'' fixed at 10. In Number Theory , the concept of P-adic Number s is also closely related to that of a power series. EXAMPLES Any Polynomial can be easily expressed as a power series around any center ''c'', albeit one with most coefficients equal to zero. For instance, the polynomial $f(x)\; =\; x^2\; +\; 2x\; +\; 3$ can be written as a power series around the center $c=0$ as ::$f(x)\; =\; 3\; +\; 2\; x\; +\; 1\; x^2\; +\; 0\; x^3\; +\; 0\; x^4\; +\; \backslash cdots\; \backslash ,$ or around the center $c=1$ as ::$f(x)\; =\; 6\; +\; 4\; (x-1)\; +\; 1(x-1)^2\; +\; 0(x-1)^3\; +\; 0(x-1)^4\; +\; \backslash cdots\; \backslash ,$ or indeed around any other center ''c''. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials. The Geometric Series formula ::
	A Power Series Will Converge For Some Values Of The Variable ''x'' (at Least For ''x''	''c'') and may diverge for others It turns out that there is always a number ''r'' with 0 &le ''r'' &le &infin such that the series converges whenever ''x'' &minus ''c'' < ''r'' and diverges whenever ''x'' &minus ''c'' > ''r'' The number ''r'' is called the ''' Radius Of Convergence ''' of the power series in general it is given as

  The Series "http://wwwinformationdelightinfo/encyclopedia/entry/absolute_convergence" class="copylinks">Converges Absolutely for ''x'' - ''c'' < ''r'' and Converges Uniformly on every Compact Subset of {''x'' : ''x'' &minus ''c'' < ''r''}
  For ''x'' - ''c'' ''r'', we cannot make any general statement on whether the series converges or diverges However, Abel's Theorem states that the sum of the series is continuous at ''x'' if the series converges at ''x''




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 Function s 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

::$$
a_n = rac {f^{\left( n ight)}\left( c ight)} {n!}


where $f^\{(n)\}(c)$ denotes the ''n''th derivative of ''f'' at ''c'', and $f^\{(0)\}(c)\; =\; f(c)$. 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 ''c''∈''U'' such that ''f'' ^(''n'')(''c'') = ''g'' ^(''n'')(''c'') for all ''n'' ≥ 0, then ''f''(''x'') = ''g''(''x'') for all ''x'' ∈ ''U''.

  Let &alpha Be A Multi-index For A Power Series ''f''(''x''₁, ''x''₂, &hellip, ''x''_''n'') The '''order''' Of The Power Series ''f'' Is Defined To Be The Least Value &alpha Such That ''a''_&alpha &ne 0, Or 0 If ''f'' &equiv 0 In Particular, For A Power Series ''f''(''x'') In A Single Variable ''x'', The Order Of ''f'' Is The Smallest Power Of ''x'' With A Nonzero Coefficient This Definition Readily Extends To "http://wwwinformationdelightinfo/encyclopedia/entry/Laurent_series" class="copylinks">Laurent Series