Editing Power series (section)

===Polynomial===
<!-- This section is linked from [[Complex plane]] -->

[[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.