Editing Power series (section)

==Examples==
===Polynomial===
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[[Image:Exp series.gif|right|thumb|The [[exponential function]] (in blue), and its improving approximation by the sum of the first ''n''&nbsp;+&thinsp;1 terms of its [[Maclaurin series|Maclaurin power series]] (in red). So<br> n=0 gives <math>f(x) = 1</math>,<br> n=1 <math>f(x) = 1 + x</math>,<br> n=2 <math>f(x)= 1 + x + x^2/2</math>, <br> n=3 <math>f(x)= 1 + x + x^2/2 + x^3/6</math> etcetera.]]

Every [[polynomial]] of degree {{mvar|d}} can be expressed as a power series around any center {{math|''c''}}, where all terms of degree higher than {{mvar|d}} have a coefficient of zero.<ref>{{cite book|author=Howard Levi|title=Polynomials, Power Series, and Calculus | url=https://books.google.com/books?id=AcI-AAAAIAAJ|year=1967|publisher=Van Nostrand|pages=24|author-link=Howard Levi}}</ref> For instance, the polynomial <math display="inline">f(x) = x^2 + 2x + 3</math> can be written as a power series around the center <math display="inline">c = 0</math> as
<math display="block">f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \cdots</math>
or around the center <math display="inline">c = 1</math> as
<math display="block">f(x) = 6 + 4(x - 1) + 1(x - 1)^2 + 0(x - 1)^3 + 0(x - 1)^4 + \cdots. </math>

One can view power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense.

===Geometric series, exponential function and sine===
The [[geometric series]] formula
<math display="block">\frac{1}{1 - x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots,</math>
which is valid for <math display="inline">|x| < 1</math>, is one of the most important examples of a power series, as are the [[exponential function]] formula
<math display="block">e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots</math>
and the [[Taylor_series#Trigonometric_functions|sine formula]]
<math display="block">\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots,</math>
valid for all real ''x''.  These power series are examples of [[Taylor series]] (or, more specifically, of [[Maclaurin series]]).

=== On the set of exponents ===

Negative powers are not permitted in an ordinary power series; for instance, <math display="inline">x^{-1} + 1 + x^{1} + x^{2} + \cdots</math> is not considered a power series (although it is a [[Laurent series]]). Similarly, fractional powers such as <math display="inline">x^\frac{1}{2}</math> are not permitted; fractional powers arise in [[Puiseux series]]. The coefficients <math display="inline"> a_n</math> must not depend on {{nowrap|<math display="inline">x</math>,}} thus for instance <math display="inline">\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \cdots </math> is not a power series.