Taylor Series Section 2. Definitions

<body bgcolor="#ffffff"> <hr> <center> <table border="2"> <tr valign="top"> <td><a href="node1.shtml"><img height="44" width="44" src="/UCESIcons/tools/previous.gif"></a></td> <td><a href="taylor_series.html"><img height="44" width="44" src="/UCESIcons/tools/up.gif"></a></td> <td><a href="node3.shtml"><img height="44" width="44" src="/UCESIcons/tools/next.gif"></a></td> <td><a href="contents-node2.html"><img height="44" width="44" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="copyright.html"><img height="44" width="44" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces@krellinst.org"><img height="44" width="44" src="/UCESIcons/tools/feedback.gif"></a></td> </tr> </table> </center> <hr> <!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.2//EN">  <HTML> <HEAD> <TITLE>Section 2. Definitions</TITLE> <BASE TARGET="_parent"> <META NAME="description" CONTENT="Section 2. Definitions"> <META NAME="keywords" CONTENT="taylor_series"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <LINK REL=STYLESHEET HREF="taylor_series.css"> </HEAD> <BODY bgcolor="#ffffff"> <h1>Section 2.   Definitions</h1> <p> <b>Definition:</b> Given a function <span class="nobr"><i>f</i>(<i>x</i>)</span> whose first <i>n</i> derivatives can be found, we define the <B>Taylor polynomial</B>  <span class="nobr"><i>P<sub>n</sub></i>(<i>x</i>)</span> of degree <i>n</i> for the function <span class="nobr"><i>f</i>(<i>x</i>)</span> about the value <i>c</i> as <p align="center"> <IMG WIDTH=572 HEIGHT=38 ALIGN=BOTTOM ALT="[Definition of P_n(x)]" SRC="img8.gif"> </p> <p> where <img width="45" height="23" align="absmiddle" alt="f^(n) (c)" src="img24.gif"/> denotes the  <i>n</i><sup>th</sup> derivative of the function <i>f</i> evaluated at <span class="nobr"><i>x=c</i> .</span> <P> The <b>Taylor series</b> associated with <i>f</i> about <i>c</i> is simply the Taylor polynomial  <span class="nobr"><i>P<sub>n</sub></i>(<i>x</i>)</span> for <i>f</i> about <i>c</i> where <IMG WIDTH=51 HEIGHT=9 ALIGN="absmiddle" ALT="n -> infinity" SRC="img11.gif"> . More precisely, we have <P> <IMG WIDTH=348 HEIGHT=46 ALIGN=BOTTOM ALT="[infinite series for f(x)]" SRC="img12.gif"> <P> This is an impressive equality, to say the least. However, in practice, we will not deal with power series as often as with a polynomial  <span class="nobr"><i>P<sub>n</sub></i>(<i>x</i>) ,</span> which is simply a truncated version of the power series. In that case, we will see that</p> <p> <IMG WIDTH=296 HEIGHT=15 ALIGN=BOTTOM ALT="f(x) ~ P_n(x)" SRC="img13.gif"> </p> <HR> <ADDRESS> <i>James A. Sellers<br> <a href="mailto:sellersj@math.psu.edu">sellersj@math.psu.edu</a></i> </ADDRESS> <P> </BODY> </HTML> <hr> <center> <table border="2"> <tr valign="top"> <td><a href="node1.shtml"><img height="44" width="44" src="/UCESIcons/tools/previous.gif"></a></td> <td><a href="taylor_series.html"><img height="44" width="44" src="/UCESIcons/tools/up.gif"></a></td> <td><a href="node3.shtml"><img height="44" width="44" src="/UCESIcons/tools/next.gif"></a></td> <td><a href="contents-node2.html"><img height="44" width="44" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="copyright.html"><img height="44" width="44" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces@krellinst.org"><img height="44" width="44" src="/UCESIcons/tools/feedback.gif"></a></td> </tr> </table> </center> <hr> </body>