9.3

<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> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Cálculo 21"><U><B>Cálculo de una variable</B></U></A><B> | </B><A HREF="id543.htm" TARGET="_top" TITLE="D.G. Zill"><U><B>D. G. Zill</B></U></A><B> |</B></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> <B>Ejercicios 9.4</B></div> <div> <B>Series infinitas</B></div> <div> <B>Prueba de la integral</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> <div>  <A NAME="top_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Top</FONT></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="953" BORDERCOLORLIGHT="#006666" BORDERCOLORDARK="#006666" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006666 HEIGHT= 23 WIDTH="943"><div> <B>Apuntes</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 ><NOBR><div> <A HREF="#teorema_9_4_1_1" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.4.1</U></A>  Prueba de la integral</div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_9_4_2_1" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.4.2</U></A>  Convergencia de la serie <I>p</I></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 27 WIDTH="471"><div> <FONT SIZE="4" COLOR="#009966" FACE="Georgia" ><A HREF="#definici_n" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición</U></A></FONT><A HREF="#definici_n" TITLE="9.3"><U>:</U></A> Serie <I>p</I></div> </tD> <tD VALIGN=TOP WIDTH="471"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="27" ALIGN="bottom" WIDTH="471" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 23 ><NOBR><div> <A HREF="#enunciados_1" TITLE="9.3"><U><B><IMG BORDER="0" SRC="Symb075c00b7011800000000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Enunciados de los problemas (enlaces)</B></U></A></div> </NOBR></tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC><NOBR><div> <A HREF="#lista_de_reproducci_n" TITLE="9.3"><U><B><IMG BORDER="0" SRC="Symb075c00b7011800000000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Lista de reproducción del Ejercicio 9.4</B></U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 BGCOLOR=#006666 HEIGHT= 14 WIDTH="943"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="14" ALIGN="bottom" WIDTH="943" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <div> <B> <A NAME="teorema_9_4_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Teorema 9.4.1</FONT></div> <div> <B><IMG SRC="2ef2e4c90.gif" border=0 width="740" height="201" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B> <A NAME="definici_n"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Definición</FONT></div> <div> <B><IMG SRC="2ef4ff980.gif" border=0 width="767" height="152" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B> <A NAME="teorema_9_4_2_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >eorema 9.4.2</FONT></div> <div> <B><IMG SRC="2ef303580.gif" border=0 width="515" height="88" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div> <B> <A NAME="enunciados_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Enunciados</FONT></div> <div> <B>Enunciados </B><B> y enlaces a  las</B><B> soluciones de los problemas de los Ejercicios 9.4</B></div> <bR> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >olución en imagen <IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> de los problemas de los Ejercicios 9.4:</FONT></div> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas </FONT><B>1</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>30</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, determine si la serie dada converge o diverge. Recurra a la prueba de la integral en los casos en que sea apropiado.</FONT></div> <div> <IMG SRC="2ef8603b0.gif" border=0 width="96" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/1dNHIQ_lloA" TARGET="_top" TITLE="https://youtu.be/1dNHIQ_lloA"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2ef76c3b0.gif" border=0 width="108" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/1dNHIQ_lloA" TARGET="_top" TITLE="https://youtu.be/1dNHIQ_lloA"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f1cc8390.gif" border=0 width="200" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=4QDhN3Osc5s" TARGET="_top" TITLE="https://www.youtube.com/watch?v=4QDhN3Os"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f1d15390.gif" border=0 width="277" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=3zZ_j2p4wT8" TARGET="_top" TITLE="https://www.youtube.com/watch?v=3zZ_j2p4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2f1e773b0.gif" border=0 width="119" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=-nWhniLM8tg" TARGET="_top" TITLE="https://www.youtube.com/watch?v=-nWhniLM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2f1f713b0.gif" border=0 width="113" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://www.youtube.com/watch?v=cyVqRhC0_KY" TARGET="_top" TITLE="https://www.youtube.com/watch?v=cyVqRhC0"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>           <IMG SRC="2f2081430.gif" border=0 width="129" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#7" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f2373430.gif" border=0 width="115" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/OqaRXjkx7qs" TARGET="_top" TITLE="https://youtu.be/OqaRXjkx7qs"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f2574430.gif" border=0 width="116" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#9" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2f2670430.gif" border=0 width="112" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#10" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f2868430.gif" border=0 width="104" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#11" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f2a84430.gif" border=0 width="132" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#12" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f226b430.gif" border=0 width="107" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/CGTbWKPPG8U" TARGET="_top" TITLE="https://youtu.be/CGTbWKPPG8U"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f2c6b430.gif" border=0 width="107" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#14" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2f2e9b440.gif" border=0 width="155" height="68" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#15" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f308d430.gif" border=0 width="141" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#16" TITLE="9.3"><U><IMG SRC="265318180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>          <IMG SRC="2f3299430.gif" border=0 width="153" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="#17" TITLE="9.3"><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>  <A NAME="lista_de_reproducci_n"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Lista de reproducción</FONT></div> <div> <B>Lista de reproducción de los Ejercicios 9.4</B></div> <bR> <div>  <iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo0F14cFnRc_KWUVoLU_8AM_" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>  </div> <bR> <div> <B>Solución de los Ejercicios 9.4</B></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>7</div> <div> <IMG SRC="2f2180000.gif" border=0 width="896" height="512" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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="2f2480f50.gif" border=0 width="896" height="501" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="10"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>10</div> <div> <IMG SRC="2f2780fd0.gif" border=0 width="896" height="509" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <IMG SRC="2f2980b50.gif" border=0 width="896" height="437" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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="2f2ba9000.gif" border=0 width="937" height="512" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="14"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>14</div> <div> <IMG SRC="2f2d55df0.gif" border=0 width="853" height="479" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <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="2f2f80080.gif" border=0 width="896" height="520" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="16"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>16</div> <div> <IMG SRC="2f3148f50.gif" border=0 width="840" height="501" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>17</div> 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