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 9.3</B></div> <div> <B>Series infinitas</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"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Top</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 WIDTH="471"><div> <A HREF="#definición_de_serie" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición de serie</U></A></div> </tD> <tD VALIGN=TOP WIDTH="471"><div> <A HREF="#sucesión_de_sumas_parciales" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Sucesión de sumas parciales</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 27 WIDTH="471"><div> <FONT SIZE="4" COLOR="#009966" FACE="Georgia" ><A HREF="#definición_9_3_1" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Definición 9.3.1</U></A></FONT>: Serie convergente</div> </tD> <tD VALIGN=TOP WIDTH="471"><div> <A HREF="#serie_telescópica" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Serie telescópica</U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 23 WIDTH="471"><div> <A HREF="#serie_geométrica" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Serie geométrica</U></A></div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_9_3_1" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.1</U></A>: <FONT SIZE="4" COLOR="#009999" FACE="Georgia" >Suma de una serie geométrica</FONT></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 WIDTH="471"><div> <A HREF="#serie_armónica" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Serie armónica</U></A></div> </tD> <tD VALIGN=TOP WIDTH="471"><div> <A HREF="#teorema_9_3_2" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.2</U></A>: <FONT SIZE="4" COLOR="#009999" FACE="Georgia" >Condición necesaria para la convergencia</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 WIDTH="471"><div> <FONT SIZE="4" COLOR="#009966" FACE="Georgia" ><A HREF="#teorema_9_3_3" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.3</U></A>: </FONT>Prueba del término <I>n</I>-ésimo para la divergencia</div> </tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#teorema_9_3_4" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.4</U></A>:<B> </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" >Múltiplo constante de una serie</FONT></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 45 ><NOBR><div> <A HREF="#teorema_9_3_5" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.5</U></A>: <FONT SIZE="4" COLOR="#009999" FACE="Georgia" >Suma de dos series convergentes</FONT></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="471"><div> <A HREF="#teorema_9_3_6" TITLE="9.3"><U><IMG BORDER="0" SRC="Symb075c00b7011866990000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 9.3.6</U></A>: <FONT SIZE="4" COLOR="#009999" FACE="Georgia" >Suma de una serie convergente y una divergente</FONT></div> </tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 23 ><NOBR><div> <A HREF="#enunciados" 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_reproducción_" 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 9.3</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>  <A NAME="definición_de_serie"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición de serie</div> <div> <B>Definición de se</B><B>rie</B></div> <div> <B><IMG SRC="2e621b890.gif" border=0 width="795" height="137" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="sucesión_de_sumas_parciales"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Sucesión de sumas parciales</div> <div> <B>Sucesión de sumas parciales</B></div> <div> <B><IMG SRC="2e63edf70.gif" border=0 width="1005" height="247" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="definición_9_3_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Definición 9.3.1</div> <div> <B>Definición 9.3.1: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Serie convergente</B></FONT></div> <div> <B><IMG SRC="2e64b8af0.gif" border=0 width="952" height="175" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="serie_telescópica"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Serie telescópica</div> <div> <B>Serie telescópica</B></div> <div> Una serie en la cual el término general de la sucesión de sumas parciales "colapsa" hasta dos términos se denomina <B>serie telescópica</B>.</div> <div>  <A NAME="serie_geométrica"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Serie geométrica</div> <div> <B>Serie geométrica</B></div> <div> Una serie donde cada término después del primero se obtiene al multiplicar el término presedente por un número real fijo <I>r</I>, se denomina <B>serie geométrica</B>:</div> <div> <B><IMG SRC="2e65674f0.gif" border=0 width="359" height="79" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_9_3_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.1</div> <div> <B>Teorema 9.3.1: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Suma de una serie geométrica</B></FONT></div> <div> <B><IMG SRC="2e6824710.gif" border=0 width="548" height="113" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B> <A NAME="serie_armónica"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Serie armónica</B></div> <div> <B>Serie armónica</B></div> <div> La <B>serie armónica</B> es la suma de los recíprocos de los enteros positivos:</div> <div> <IMG SRC="2e6c15310.gif" border=0 width="277" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> La serie armónica es  <B>divergente</B>.</div> <div>  <A NAME="teorema_9_3_2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.2</div> <div> <B>Teorema 9.3.2: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Condición necesaria para la convergencia</B></FONT></div> <div> <B><IMG SRC="2e6d9f270.gif" border=0 width="415" height="39" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_9_3_3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.23</div> <div> <B>Teorema 9.3.</B><B>3</B><B>: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Prueba del término </B><I><B>n</B></I><B>-ésimo para la divergencia</B></FONT></div> <div> <B><IMG SRC="2e6e8f270.gif" border=0 width="399" height="39" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_9_3_4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.4</div> <div> <B>Teorema 9.3.4: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Múltiplo constante de una serie</B></FONT></div> <div> <B><IMG SRC="2e71f71f0.gif" border=0 width="1015" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_9_3_5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.5</div> <div> <B>Teorema 9.3.5: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Suma de dos series convergentes</B></FONT></div> <div> <B><IMG SRC="2e703c630.gif" border=0 width="572" height="99" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div>  <A NAME="teorema_9_3_6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 9.3.6</div> <div> <B>Teorema 9.3.6</B><B>: </B><FONT SIZE="4" COLOR="#009999" FACE="Georgia" ><B>Suma de una serie convergente y una divergente</B></FONT></div> <div> <B><IMG SRC="2e6f4c1f0.gif" border=0 width="588" height="31" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="enunciados"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Enunciados</div> <div> <B>Enunciados y soluciones de los problemas de los Ejercicios 9.3</B></div> <bR> <div> <FONT SIZE="5" COLOR="#006600" FACE="Bell MT" ><B>S</B></FONT>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 9.3:</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>10</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, escriba los primeros cuatro términos de cada serie.</FONT></div> <div> <IMG SRC="2e375d390.gif" border=0 width="93" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/1os_koCbdh0" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e3848390.gif" border=0 width="72" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/1os_koCbdh0" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e396d3f0.gif" border=0 width="109" height="63" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/1os_koCbdh0" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e3f6d3f0.gif" border=0 width="109" height="63" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/nYhk7YEx_RY" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4055390.gif" border=0 width="85" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/nYhk7YEx_RY" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e415f3b0.gif" border=0 width="95" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/nYhk7YEx_RY" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2e42ac3b0.gif" border=0 width="172" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/HBGuBcIbMzI" TARGET="_top" TITLE="https://youtu.be/HBGuBcIbMzI"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e43ac3b0.gif" border=0 width="172" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/HBGuBcIbMzI" TARGET="_top" TITLE="https://youtu.be/HBGuBcIbMzI"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e44653c0.gif" border=0 width="101" height="60" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qqUwC99LrkU" TARGET="_top" TITLE="https://youtu.be/qqUwC99LrkU"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4571390.gif" border=0 width="113" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qqUwC99LrkU" TARGET="_top" TITLE="https://youtu.be/qqUwC99LrkU"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></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>11</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>14</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, encuentre la suma de la serie telescópica dada.</FONT></div> <div> <IMG SRC="2e4674340.gif" border=0 width="116" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/lnHkub9XYwU" TARGET="_top" TITLE="https://youtu.be/lnHkub9XYwU"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e479c340.gif" border=0 width="156" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/Q_UisXKezRM" TARGET="_top" TITLE="https://youtu.be/Q_UisXKezRM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>    <IMG SRC="2e486d310.gif" border=0 width="109" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/AObo1H3rWCg" TARGET="_top" TITLE="https://youtu.be/AObo1H3rWCg"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4995310.gif" border=0 width="149" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/uDLyr8TNp_Y" TARGET="_top" TITLE="https://youtu.be/uDLyr8TNp_Y"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></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>15</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>2</B><B>4</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, determine si la serie geométrica dada converge o diverge. Si es convergente, encuentre la suma de la serie.</FONT></div> <div> <IMG SRC="2e4a783b0.gif" border=0 width="120" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/BvIOtqdcHJI" TARGET="_top" TITLE="https://youtu.be/BvIOtqdcHJI"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4b853b0.gif" border=0 width="133" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/ivGaNsYqzU4" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4c65390.gif" border=0 width="101" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/ivGaNsYqzU4" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4e83390.gif" border=0 width="131" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/ivGaNsYqzU4" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e4f8f390.gif" border=0 width="143" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qFYV_hiPe_w" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2e506d3d0.gif" border=0 width="109" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qFYV_hiPe_w" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5195440.gif" border=0 width="149" height="68" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qFYV_hiPe_w" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e528f400.gif" border=0 width="143" height="64" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/qFYV_hiPe_w" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       </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>25</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>30</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, escriba cada número decimal que se repite como un cociente de enteros.</FONT></div> <div> <IMG SRC="2e536b140.gif" border=0 width="107" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/0CSLHXqbtq8" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <IMG SRC="2e5464140.gif" border=0 width="100" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/0CSLHXqbtq8" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5583140.gif" border=0 width="131" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/0CSLHXqbtq8" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5684140.gif" border=0 width="132" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/dGdOyPRzoyM" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e57a1140.gif" border=0 width="161" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/dGdOyPRzoyM" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2e588d140.gif" border=0 width="141" height="20" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/dGdOyPRzoyM" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></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>31</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >y</FONT> <B>32</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, escriba cada número decimal que se repite como un cociente de enteros.</FONT></div> <div> <IMG SRC="2e59c33f0.gif" border=0 width="195" height="63" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/JUjT0SJrqX4" TARGET="_top" TITLE="https://youtu.be/JUjT0SJrqX4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5a78370.gif" border=0 width="120" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/JUjT0SJrqX4" TARGET="_top" TITLE="https://youtu.be/JUjT0SJrqX4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></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>33</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >a</FONT> <B>42</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, muestre que la serie dada es divergente.</FONT></div> <div> <IMG SRC="2e5b4b310.gif" border=0 width="75" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/D5QMjKnRysg" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5c70310.gif" border=0 width="112" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/D5QMjKnRysg" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5d67310.gif" border=0 width="103" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/D5QMjKnRysg" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5e8d340.gif" border=0 width="141" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/D5QMjKnRysg" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e5f63310.gif" border=0 width="99" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/f1mXH6Ts-U4" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e6088330.gif" border=0 width="136" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/f1mXH6Ts-U4" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="2e614d310.gif" border=0 width="77" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/f1mXH6Ts-U4" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e6950310.gif" border=0 width="80" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/I5ivybcdIzE" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e6a8d330.gif" border=0 width="141" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/I5ivybcdIzE" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e6b6f310.gif" border=0 width="111" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/I5ivybcdIzE" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#006666" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" ><B>43</B> </FONT>a<FONT SIZE="4" COLOR="#CC0033" FACE="Bell MT" > <B>46</B></FONT>, determine los valores de <I>x</I> para los cuales la serie dada converge.</div> <div> <IMG SRC="2e726f370.gif" border=0 width="111" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/Oif4k8Nr3ug" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e736f370.gif" border=0 width="111" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/Oif4k8Nr3ug" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e7473310.gif" border=0 width="115" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/Oif4k8Nr3ug" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <IMG SRC="2e756b310.gif" border=0 width="107" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/Oif4k8Nr3ug" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" >47.</FONT><FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" > </FONT>Se deja caer una pelota desde una altura inicial de 15 pies sobre una plancha de concreto. </div> <div> Cada vez que la pelota rebota alcanza una altura de 2/3 de su altura precedente. </div> <div> Determine la distancia que la pelota recorre antes de quedar en reposo.<A HREF="https://youtu.be/RveNlclZz7w" TARGET="_top" TITLE="https://youtu.be/RveNlclZz7w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" >48</FONT><FONT SIZE="4" COLOR="#CC0033" FACE="Times New Roman" >. </FONT>En el problema 47 determine el tiempo que tarda la pelota en llegar al reposo.<A HREF="https://youtu.be/RveNlclZz7w" TARGET="_top" TITLE="https://youtu.be/RveNlclZz7w"><U><IMG SRC="265418180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <IMG SRC="2e7be99c0.gif" border=0 width="1001" height="156" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/WSugxwuRA2U" TARGET="_top" TITLE="https://youtu.be/WSugxwuRA2U"><U><IMG SRC="265418180.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_reproducción_"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Lista de reproducción </div> <div> <B>Lista de reproducción de los Ejercicios 9.3</B></div> <bR> <div>  <iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo16ivPq9c6tSH3QA-YPwTdx" 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