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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1">PREPOWER:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:36"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E5:36"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
(  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a>  is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i>, <a class="ref" href="series_1.html#T28">SERIES_1:28</a></i>;<br/>
<b>hence </b> <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="prepower.html#K2">GeoSeq</a> </span>)</span> <a href="seqm_3.html#K1">^\</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="series_1.html#T13">SERIES_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r</font></span>)</span>

;<br/><br/>


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