Series

<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="#003366" TEXT="#F7F7FF"> <div> <B>| </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>MatU</B></U></A><B>  | </B><A HREF="id82.htm" TARGET="_top" TITLE="Sucesiones y series"><U><B>Sucesiones y s</B></U></A><A HREF="id82.htm" TARGET="_top" TITLE="Sucesiones y series"><U><B>eries</B></U></A><B> |</B></div> </td> </tr><tr> <td valign="top" rowspan="2" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div>  <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-6266400581769473" data-ad-slot="5543838917"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> </td> <td valign="top" height=442 colspan=2 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>S e r i e s</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-6266400581769473" data-ad-slot="6260336925" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>Enunciados de los ejercicios y problemas </B></div> <div> <B>M i s c e l á n e a</B></div> <div> (Pulsa sobre el enunciado del ejercicio o problema de tú interés para acceder a la solución del mismo, en imagen o en video)</div> <bR> <div> <B>          </B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><B>E</B></FONT><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >scriba los primeros cuatro términos de cada serie:</FONT></div> <div> <A HREF="https://youtu.be/1os_koCbdh0?t=82" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0?t=82"><U><IMG SRC="3ee5c310.gif" border=0 width="92" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/1os_koCbdh0?t=208" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0?t=208"><U><IMG SRC="3ef45340.gif" border=0 width="69" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/1os_koCbdh0?t=314" TARGET="_top" TITLE="https://youtu.be/1os_koCbdh0?t=314"><U><IMG SRC="3f06c3c0.gif" border=0 width="108" height="60" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/nYhk7YEx_RY?t=69" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY?t=69"><U><IMG SRC="3f1693b0.gif" border=0 width="105" height="59" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/nYhk7YEx_RY?t=279" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY?t=279"><U><IMG SRC="3f255310.gif" border=0 width="85" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/nYhk7YEx_RY?t=435" TARGET="_top" TITLE="https://youtu.be/nYhk7YEx_RY?t=435"><U><IMG SRC="3f35c350.gif" border=0 width="92" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/HBGuBcIbMzI?t=79" TARGET="_top" TITLE="https://youtu.be/HBGuBcIbMzI?t=79"><U><IMG SRC="3f4b0370.gif" border=0 width="176" height="55" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A> </div> <div> <A HREF="https://youtu.be/HBGuBcIbMzI?t=341" TARGET="_top" TITLE="https://youtu.be/HBGuBcIbMzI?t=341"><U><IMG SRC="3f5b0350.gif" border=0 width="176" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/qqUwC99LrkU?t=77" TARGET="_top" TITLE="https://youtu.be/qqUwC99LrkU?t=77"><U><IMG SRC="3f665350.gif" border=0 width="101" height="53" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/qqUwC99LrkU?t=282" TARGET="_top" TITLE="https://youtu.be/qqUwC99LrkU?t=282"><U><IMG SRC="3f76f310.gif" border=0 width="111" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div>           <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><B>E</B></FONT>ncuentre la suma de la serie telescópica dada:</div> <div> <A HREF="https://youtu.be/lnHkub9XYwU" TARGET="_top" TITLE="https://youtu.be/lnHkub9XYwU"><U><IMG SRC="3f875330.gif" border=0 width="117" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/Q_UisXKezRM" TARGET="_top" TITLE="https://youtu.be/Q_UisXKezRM"><U><IMG SRC="3f99f330.gif" border=0 width="159" height="51" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/AObo1H3rWCg" TARGET="_top" TITLE="https://youtu.be/AObo1H3rWCg"><U><IMG SRC="3fa6f310.gif" border=0 width="111" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/uDLyr8TNp_Y" TARGET="_top" TITLE="https://youtu.be/uDLyr8TNp_Y"><U><IMG SRC="3fb98310.gif" border=0 width="152" height="49" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div>           <B>D</B>etermine si la serie geométrica dada converge o diverge. Si es convergente, encuentre la suma de la serie:</div> <div> <A HREF="https://youtu.be/BvIOtqdcHJI?t=110" TARGET="_top" TITLE="https://youtu.be/BvIOtqdcHJI?t=110"><U><IMG SRC="3fc7f3d0.gif" border=0 width="127" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/BvIOtqdcHJI?t=256" TARGET="_top" TITLE="https://youtu.be/BvIOtqdcHJI?t=256"><U><IMG SRC="3fd8b3d0.gif" border=0 width="139" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/BvIOtqdcHJI?t=405" TARGET="_top" TITLE="https://youtu.be/BvIOtqdcHJI?t=405"><U><IMG SRC="3fe7c410.gif" border=0 width="124" height="65" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/ivGaNsYqzU4?t=111" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4?t=111"><U><IMG SRC="3ff8d3d0.gif" border=0 width="141" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/ivGaNsYqzU4?t=292" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4?t=292"><U><IMG SRC="4006b380.gif" border=0 width="107" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/ivGaNsYqzU4?t=435" TARGET="_top" TITLE="https://youtu.be/ivGaNsYqzU4?t=435"><U><IMG SRC="4018c380.gif" border=0 width="140" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="https://youtu.be/qFYV_hiPe_w?t=109" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w?t=109"><U><IMG SRC="4029c380.gif" border=0 width="156" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/qFYV_hiPe_w?t=286" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w?t=286"><U><IMG SRC="40375410.gif" border=0 width="117" height="65" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <A HREF="https://youtu.be/qFYV_hiPe_w?t=380" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w?t=380"><U><IMG SRC="404a5480.gif" border=0 width="165" height="72" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/qFYV_hiPe_w?t=572" TARGET="_top" TITLE="https://youtu.be/qFYV_hiPe_w?t=572"><U><IMG SRC="40597440.gif" border=0 width="151" height="68" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <B>          E</B>scriba cada número decimal que se repite como un cociente de enteros:</div> <div> <A HREF="https://youtu.be/0CSLHXqbtq8?t=109" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8?t=109"><U><IMG SRC="4067b150.gif" border=0 width="123" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <A HREF="https://youtu.be/0CSLHXqbtq8?t=279" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8?t=279"><U><IMG SRC="40771150.gif" border=0 width="113" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/0CSLHXqbtq8?t=459" TARGET="_top" TITLE="https://youtu.be/0CSLHXqbtq8?t=459"><U><IMG SRC="40894150.gif" border=0 width="148" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/dGdOyPRzoyM?t=110" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM?t=110"><U><IMG SRC="40997150.gif" border=0 width="151" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/dGdOyPRzoyM?t=396" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM?t=396"><U><IMG SRC="40a94150.gif" border=0 width="148" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>        <A HREF="https://youtu.be/dGdOyPRzoyM?t=661" TARGET="_top" TITLE="https://youtu.be/dGdOyPRzoyM?t=661"><U><IMG SRC="40ba1150.gif" border=0 width="161" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <B>          E</B>ncuentre la suma de las series dadas:</div> <div> <A HREF="https://youtu.be/JUjT0SJrqX4?t=25" TARGET="_top" TITLE="https://youtu.be/JUjT0SJrqX4?t=25"><U><IMG SRC="40cd4430.gif" border=0 width="212" height="67" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/JUjT0SJrqX4?t=277" TARGET="_top" TITLE="https://youtu.be/JUjT0SJrqX4?t=277"><U><IMG SRC="40d843d0.gif" border=0 width="132" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <B>          M</B>uestre que la serie dada es divergente:</div> <div> <A HREF="https://youtu.be/D5QMjKnRysg?t=142" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg?t=142"><U><IMG SRC="40e51380.gif" border=0 width="81" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/D5QMjKnRysg?t=194" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg?t=194"><U><IMG SRC="40f7c380.gif" border=0 width="124" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/D5QMjKnRysg?t=244" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg?t=244"><U><IMG SRC="41071380.gif" border=0 width="113" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/D5QMjKnRysg?t=393" TARGET="_top" TITLE="https://youtu.be/D5QMjKnRysg?t=393"><U><IMG SRC="4119d390.gif" border=0 width="157" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/f1mXH6Ts-U4?t=138" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4?t=138"><U><IMG SRC="4126d380.gif" border=0 width="109" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/f1mXH6Ts-U4?t=217" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4?t=217"><U><IMG SRC="41398390.gif" border=0 width="152" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <A HREF="https://youtu.be/f1mXH6Ts-U4?t=365" TARGET="_top" TITLE="https://youtu.be/f1mXH6Ts-U4?t=365"><U><IMG SRC="41455380.gif" border=0 width="85" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <A HREF="https://youtu.be/I5ivybcdIzE?t=27" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE?t=27"><U><IMG SRC="41557380.gif" border=0 width="87" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>      <A HREF="https://youtu.be/I5ivybcdIzE?t=158" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE?t=158"><U><IMG SRC="41699390.gif" border=0 width="153" height="57" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/I5ivybcdIzE?t=350" TARGET="_top" TITLE="https://youtu.be/I5ivybcdIzE?t=350"><U><IMG SRC="4177b380.gif" border=0 width="123" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <B>          D</B>etermine los valores de <I>x</I> para los cuales la serie dada converge:</div> <div> <A HREF="https://youtu.be/Oif4k8Nr3ug?t=61" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug?t=61"><U><IMG SRC="418773d0.gif" border=0 width="119" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/Oif4k8Nr3ug?t=166" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug?t=166"><U><IMG SRC="419773d0.gif" border=0 width="119" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/Oif4k8Nr3ug?t=316" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug?t=316"><U><IMG SRC="41a7d380.gif" border=0 width="125" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       <A HREF="https://youtu.be/Oif4k8Nr3ug?t=472" TARGET="_top" TITLE="https://youtu.be/Oif4k8Nr3ug?t=472"><U><IMG SRC="41b74380.gif" border=0 width="116" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >47.</FONT> Se deja caer una pelota desde una altura inicial de 15 pies sobre una plancha de concreto. Cada vez que la pelota rebota, alcanza una altura de de su altura precedente. Recurra a la serie geométrica para determinar la distancia que la pelota recorre antes de quedar en reposo.  <A HREF="https://youtu.be/RveNlclZz7w?t=38" TARGET="_top" TITLE="https://youtu.be/RveNlclZz7w?t=38"><U>Ver la solución en video</U></A></div> <bR> <div> <FONT SIZE="4" COLOR="#CC0033" FACE="Georgia" >48.</FONT> En el problema 47 determine el tiempo que tarda la pelo?ta en llegar al reposo.  <A HREF="https://youtu.be/RveNlclZz7w?t=451" TARGET="_top" TITLE="https://youtu.be/RveNlclZz7w?t=451"><U>Ver la solución en video</U></A></div> <bR> <div> <A HREF="https://youtu.be/WSugxwuRA2U?t=26" TARGET="_top" TITLE="https://youtu.be/WSugxwuRA2U?t=26"><U><IMG SRC="41c93ef0.gif" border=0 width="915" height="239" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://youtu.be/WSugxwuRA2U?t=26" TARGET="_top" TITLE="https://youtu.be/WSugxwuRA2U?t=26"><U>Ver la solución</U></A></div> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="9294650114"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> </td> </tr><tr> <td valign="top" colspan=2 height=14 BGCOLOR="#003366" TEXT="#F7F7FF"> <bR> </td> </tr></table></body>