8.5

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B>C á l c u l o  2 1</B><FONT SIZE="4" COLOR="#FFFFCC" FACE="Bell MT" >   </FONT><B>  | </B>  <A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U>Home</U></A>  |  <A HREF="id98.htm" TARGET="_top" TITLE="Contenido"><U>Contenido</U></A>  | <A HREF="id488.htm" TARGET="_top" TITLE="Leithold EC7"><U>Leithold</U></A> </div> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> Ejercicios 8.5</div> <div> <B>Series infinitas de términos positivos y negativos</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3926688482" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B> <A NAME="criterio_de_las_series_alternantes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Criterio de las series alternantes</B></div> <div> <IMG SRC="276d2fe30.gif" border=0 width="303" height="227" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_de_la_cota_superior_del_error"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema de la cota superior del error en la aproximación de una serie</B></div> <div> <IMG SRC="276eab0a0.gif" border=0 width="427" height="266" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_criterio_de_la_razón"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema: Criterio de la razón</B></div> <div> <IMG SRC="276fae5a0.gif" border=0 width="430" height="346" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B> <A NAME="teorema_criterio_de_la_raíz"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Teorema: Criterio de la raíz</B></div> <div> <B><IMG SRC="2770a6310.gif" border=0 width="422" height="305" ALT="Imagen de Paintbrush" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >1</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >14</FONT>, determine si la <A HREF="#criterio_de_las_series_alternantes" TITLE="8.5"><U>serie alternante </U></A>es convergente o divergente.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="277385300.gif" border=0 width="133" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="277885300.gif" border=0 width="133" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#3" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="277994300.gif" border=0 width="148" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#6" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="277a8f300.gif" border=0 width="143" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#9" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="277b93330.gif" border=0 width="147" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#12" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >15</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >22</FONT>, obtenga una cota superior para el error si la suma de los primeros cuatro términos se emplea como una aproximación de la suma de la serie infinita.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="277d83300.gif" border=0 width="131" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#15" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="277fa9300.gif" border=0 width="169" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#18" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2781dc300.gif" border=0 width="220" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#21" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >23</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >28</FONT>, calcule la suma de la serie infinita, con una exactitud de tres cifras decimales.</div> <div> <IMG SRC="27858c300.gif" border=0 width="140" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#24" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="278788300.gif" border=0 width="136" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#25" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Times New Roman" ><B>E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >29</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >48</FONT>, determine si la serie es absolutamente convergente, condicionalmente convergente o divergente. Justifique la respuesta.</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Times New Roman" ><IMG SRC="278989300.gif" border=0 width="137" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#35" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <IMG SRC="278b7f490.gif" border=0 width="127" height="73" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#46" TITLE="8.5"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> Soluciones</div> <bR> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <IMG SRC="2771fb2c0.gif" border=0 width="763" height="300" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3</div> <div> <IMG SRC="2774fb330.gif" border=0 width="763" height="307" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <DIV ALIGN="LEFT"></DIV> <div> Arriba</div> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>6</div> <div> <IMG SRC="2775d1300.gif" border=0 width="721" height="304" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="9"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>9</div> <div> <IMG SRC="27764fad0.gif" border=0 width="847" height="429" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>12</div> <div> <IMG SRC="27774f2c0.gif" border=0 width="847" height="556" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="15"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>15</div> <div> <IMG SRC="277cb74c0.gif" border=0 width="695" height="332" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="18"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>18</div> <div> <IMG SRC="277e98630.gif" border=0 width="664" height="355" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>21</div> <div> <IMG SRC="2780e35c0.gif" border=0 width="739" height="348" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="24"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>24</div> <div> <IMG SRC="2782fc280.gif" border=0 width="764" height="552" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>25</div> <div> <IMG SRC="2786d5f30.gif" border=0 width="725" height="499" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="35"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>35</div> <div> <IMG SRC="2788b88b0.gif" border=0 width="696" height="395" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="46"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>46</div> <div> <IMG SRC="278a9f010.gif" border=0 width="671" height="513" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> </td> <td valign="top" rowspan="2" width=105 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><B><IMG SRC="2c0564580.gif" border=0 width="100" height="600" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <bR> </td> </tr><tr> <td valign="top" height=24 BGCOLOR="#006666" TEXT="#080000"> <div> <B> <form action="http://www.google.com.co" id="cse-search-box" target="_blank"> <div> <input type="hidden" name="cx" value="partner-pub-6266400581769473:4577786115" /> <input type="hidden" name="ie" value="UTF-8" /> <input type="text" name="q" size="100" /> <input type="submit" name="sa" value="Buscar" /> </div> </form> <script type="text/javascript" src="http://www.google.com.co/coop/cse/brand?form=cse-search-box&lang=es"></script>  </B></div> </td> </tr></table></body>