document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=236799>Question 236799</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the following series is divergent?\r<BR>\n" );
document.write( "\n" );
document.write( "1 + 3(1/4) + 9(1/4)^2 + 27(1/4)^3 +...\r<BR>\n" );
document.write( "\n" );
document.write( "1 + 3(1/5) + 9(1/5)^2 + 27(1/5)^3 +...\r<BR>\n" );
document.write( "\n" );
document.write( "1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +...  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174238 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=174238\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first one to get you started.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "# 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take note that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1+3\left(\frac{1}{4}\right)+9\left(\frac{1}{4}\right)^2+27\left(\frac{1}{4}\right)^3+\cdots=1+\left(\frac{3}{4}\right)^1+\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is a geometric series generated by the sequence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_{n}=a r^{n-1}\"> for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n \ge 1\">. In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3%2F4 ALIGN=MIDDLE  ALT=\"r=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now recall that an infinite geometric series only converges if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29=abs%283%2F4%29=3%2F4=0.75%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29=abs%283%2F4%29=3%2F4=0.75%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > holds, this means that this infinite geometric series converges.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1+3\left(\frac{1}{4}\right)+9\left(\frac{1}{4}\right)^2+27\left(\frac{1}{4}\right)^3+\cdots\"> adds up to some finite number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "# 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take note that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1+3\left(\frac{1}{5}\right)+9\left(\frac{1}{5}\right)^2+27\left(\frac{1}{5}\right)^3+\cdots=1+\left(\frac{3}{5}\right)^1+\left(\frac{3}{5}\right)^2+\left(\frac{3}{5}\right)^3+\cdots\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is a geometric series generated by the sequence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_{n}=a r^{n-1}\"> for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n \ge 1\">. In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3%2F5 ALIGN=MIDDLE  ALT=\"r=3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now recall that an infinite geometric series only converges if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29=abs%283%2F5%29=3%2F5=0.6%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29=abs%283%2F5%29=3%2F5=0.6%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > holds, this means that this infinite geometric series converges.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1+3\left(\frac{1}{5}\right)+9\left(\frac{1}{5}\right)^2+27\left(\frac{1}{5}\right)^3+\cdots\"> adds up to some constant.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "# 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take note that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1+3\left(\frac{1}{2}\right)+9\left(\frac{1}{2}\right)^2+27\left(\frac{1}{2}\right)^3+\cdots=1+\left(\frac{3}{2}\right)^1+\left(\frac{3}{2}\right)^2+\left(\frac{3}{2}\right)^3+\cdots\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is a geometric series generated by the sequence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_{n}=a r^{n-1}\"> for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n \ge 1\">. In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3%2F2 ALIGN=MIDDLE  ALT=\"r=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now recall that an infinite geometric series only converges if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29=abs%283%2F2%29=3%2F2=1.5%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29=abs%283%2F2%29=3%2F2=1.5%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT true, this means that this infinite geometric does NOT converge. So the series diverges. In other words, the infinite series does not add up to some constant number. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );