<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:75"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:75"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i><a class="ref" href="comseq_3.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:75"/><b>then </b>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E4:75"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E5:75"/><i><font color="Green">E49</font></i>: 
(  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E6:75"/>
(  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="comseq_3.html#E2:75"><i><font color="Green">E47</font></i></a>, <a class="ref" href="comseq_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E7:75"/><b>then </b>
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E8:75"/>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="comseq_3.html#K8" target="_self">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p3">(<span class="default"><a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E4:75"><i><font color="Green">E48</font></i></a>, <a class="txt" href="comseq_3.html#E5:75"><i><font color="Green">E49</font></i></a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a>, <a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>


</div>
