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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">s</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:12"/>
<font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:12"/><b>then </b>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>
<a NAME="E4:12"/><b>then </b>
 <a href="series_1.html#K1" target="_self">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font> is <a href="series_1.html#V1" target="_self">summable</a>
 <b>by </b><i/>;<br/>

<b>thus </b> <a href="series_1.html#K2" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="seq_1.html#K14">(#)</a> <span class="p2">(<span class="default"><a href="series_1.html#K1" target="_self">Partial_Sums</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" target="_self">Sum</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
;<br/>


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