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<HTML><HEAD><TITLE>About Geometric Series Summation</TITLE></HEAD><BODY bgcolor="#FFFFFF"> <a href="../index.html" target="_top"><img src="../icons/usins.gif" alt="Logo" align=right></a><b>Data Structures and Algorithms with Object-Oriented Design Patterns in Python</b><br><A NAME="tex2html1816" HREF="page54.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1814" HREF="page49.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1808" HREF="page52.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1818" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H2><A NAME="SECTION002240000000000000000">About Geometric Series Summation</A></H2><P><A NAME="secmodelaboutgeometric">&#160;</A><P>The series,  <IMG WIDTH=92 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline58173" SRC="img152.gif"  >,is a <em>geometric series</em><A NAME=1003>&#160;</A>and the summation<P> <IMG WIDTH=288 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath58171" SRC="img153.gif"  ><P>is called the<em>geometric series summation</em><A NAME=1007>&#160;</A>.<P>The summation can be solved as follows:First, we make the simple variable substitution <I>i</I>=<I>j</I>-1:<P><A NAME="eqnmodelgeom">&#160;</A> <IMG WIDTH=500 HEIGHT=169 ALIGN=BOTTOM ALT="eqnarray1008" SRC="img154.gif"  ><P>Note that the summation which appears on the rightis identical to the left hand side.Rearranging Equation&nbsp;<A HREF="page53.html#eqnmodelgeom"><IMG  ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>, and simplifying gives<P><A NAME="eqngeometric">&#160;</A> <IMG WIDTH=500 HEIGHT=44 ALIGN=BOTTOM ALT="equation1026" SRC="img155.gif"  ><P><HR><A NAME="tex2html1816" HREF="page54.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1814" HREF="page49.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1808" HREF="page52.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1818" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <P><ADDRESS><img src="../icons/bruno.gif" alt="Bruno" align=right><a href="../copyright.html">Copyright &#169; 2003</a> by <a href="../signature.html">Bruno R. Preiss, P.Eng.</a>  All rights reserved.</ADDRESS></BODY></HTML>

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