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

<b>assume </b><a NAME="E1:73"/>
<font color="Maroon">seq</font> is <a href="comseq_3.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:73"/><b>then </b>
 <a href="comseq_3.html#K7" target="_self">Partial_Sums</a> <font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:73"/><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">seq</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">seq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i><a class="ref" href="comseq_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:73"/><b>then </b>
(  <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">seq</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">seq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:73"/><b>then </b>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font> is <a href="series_1.html#V1">summable</a> )
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E6:73"/><b>then </b>
(  <a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K3" target="_self">Re</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="comseq_3.html#K4" target="_self">Im</a> <font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a> &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> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="series_1.html#T7">SERIES_1:7</a></i>;<br/>
<a NAME="E7:73"/><b>then </b>
( <font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:73"/>
( <font color="Maroon">seq</font> is <a href="comseq_2.html#V2">convergent</a> &amp;  <a href="comseq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a>  )
 <b>by </b><i>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>


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