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<div><span class="kw">theorem </span><a NAME="T41"><span class="comment"><font color="firebrick">:: SERIES_1:41</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K56" title="VALUED_1:func.56">abs</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) &amp;  ex <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/> for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">n</font> holds <br/><font color="Olive" title="b2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> not <font color="Olive" title="b1">s</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a> </div></div>
