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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c1">k</font> = 
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="irrat_1.html#D5" target="_self" title="IRRAT_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="newton.html#K12" title="NEWTON:func.12">!</a> </span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D9" title="SIN_COS:def.9">SIN_COS:def 9</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E2:30"/><b>then </b><i><font color="Green" title="E28">A1</font></i>: 
 <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:30"/>
 <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a>  is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="sin_cos.html#T49" title="SIN_COS:th.49">SIN_COS:49</a></i>;<br/>

<b>thus </b> <a href="sin_cos.html#K29" title="SIN_COS:func.29">exp_R</a> 1 = 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
<b>by </b><i><a class="ref" href="sin_cos.html#D27" title="SIN_COS:def.27">SIN_COS:def 27</a></i>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <a href="irrat_1.html#K5" title="IRRAT_1:func.5">eseq</a> 
<b>by </b><i><a class="txt" href="irrat_1.html#E2:30"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="sin_cos.html#D26" title="SIN_COS:def.26">SIN_COS:def 26</a></i>
;<br/>


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