Editing Taylor's theorem (section)

=== Example ===

[[File:Function with two poles.png|thumb|right|Complex plot of <math display="inline">f(z)=\frac{1}{1+z^2}</math>. Modulus is shown by elevation and argument by coloring: cyan&nbsp;=&nbsp;<math display="inline">0</math>, blue&nbsp;=&nbsp;<math display="inline">\frac{\pi}{3}</math>, violet&nbsp;=&nbsp;<math display="inline">\frac{2\pi}{3}</math>, red&nbsp;=&nbsp;<math>\pi</math>, yellow&nbsp;=&nbsp;<math display="inline">\frac{4\pi}{3}</math>, green&nbsp;=&nbsp;<math display="inline">\frac{5\pi}{3}</math>.]]
The function

<math display="block">\begin{align}
& f : \R \to \R \\
& f(x) = \frac{1}{1+x^2}
\end{align}</math>

is [[analytic function|real analytic]], that is, locally determined by its Taylor series. This function was plotted [[Taylor's theorem#Motivation|above]] to illustrate the fact that some elementary functions cannot be approximated by Taylor polynomials in neighborhoods of the center of expansion which are too large. This kind of behavior is easily understood in the framework of complex analysis. Namely, the function ''f'' extends into a [[meromorphic function]]

<math display="block">\begin{align}
& f:\Complex \cup \{\infty\} \to \Complex \cup \{\infty\} \\
& f(z) = \frac{1}{1+z^2}
\end{align}</math>

on the compactified complex plane. It has simple poles at <math display="inline">z=i</math> and <math display="inline">z=-i</math>, and it is analytic elsewhere. Now its Taylor series centered at ''z''<sub>0</sub> converges on any disc ''B''(''z''<sub>0</sub>, ''r'') with ''r'' < |''z''&nbsp;−&nbsp;''z''<sub>0</sub>|, where the same Taylor series converges at ''z''&nbsp;∈&nbsp;'''C'''. Therefore, Taylor series of ''f'' centered at 0 converges on ''B''(0, 1) and it does not converge for any ''z'' ∈ '''C''' with |''z''|&nbsp;>&nbsp;1 due to the poles at ''i'' and −''i''. For the same reason the Taylor series of ''f'' centered at 1 converges on <math display="inline">B(1, \sqrt{2})</math> and does not converge for any ''z''&nbsp;∈&nbsp;'''C''' with <math display="inline">\left\vert z-1 \right\vert>\sqrt{2}</math>.