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<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/>

<a NAME="E1:44"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">n</font> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/>
<a NAME="E2:44"/><i><font color="Green">E41</font></i>: 
<a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i>, <a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
<a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="newton.html#K12">!</a> </span>)</span>
<b>by </b><i/>
;<br/><a NAME="E2:44_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="newton.html#K12">!</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T23">NEWTON:23</a></i>;<br/><b>hence </b><a NAME="E3:44_1"/>
<span class="p1">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="irrat_1.html#E2:44"><i><font color="Green">E41</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:44"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="irrat_1.html#E1:44"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:44"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K12">!</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="irrat_1.html#K5" target="_self">eseq</a>  <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="irrat_1.html#T1" target="_self">Th1</a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>


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