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<TITLE>About Geometric Series Summation</TITLE>
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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION003240000000000000000">About Geometric Series Summation</A></H2>
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<A NAME="secmodelaboutgeometric"> </A>
<P>
The series, <IMG WIDTH=93 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline58723" SRC="img155.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img155.gif" >,
is a <em>geometric series</em><A NAME=944> </A>
and the summation
<P> <IMG WIDTH=288 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath58721" SRC="img156.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img156.gif" ><P>
is called the
<em>geometric series summation</em><A NAME=948> </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"> </A> <IMG WIDTH=500 HEIGHT=168 ALIGN=BOTTOM ALT="eqnarray949" SRC="img157.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img157.gif" ><P>
Note that the summation which appears on the right
is identical to the left hand side.
Rearranging Equation <A HREF="page51.html#eqnmodelgeom" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page51.html#eqnmodelgeom"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>, and simplifying gives
<P><A NAME="eqngeometric"> </A> <IMG WIDTH=500 HEIGHT=43 ALIGN=BOTTOM ALT="equation967" SRC="img158.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img158.gif" ><P><HR><A NAME="tex2html2523" HREF="page52.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page52.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html2521" HREF="page47.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page47.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html2515" HREF="page50.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page50.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html2525" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html2526" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <P><ADDRESS>
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