document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1040821>Question 1040821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate:\r<BR>\n" );
document.write( "\n" );
document.write( " &#8734;<BR>\n" );
document.write( " &#931; (1/3)(-4/3)^k<BR>\n" );
document.write( "k=1\r<BR>\n" );
document.write( "\n" );
document.write( "I hope I asked this right, the infinite is on top of the sigma. K=1 is on the bottom. And the rest is to the right of the sigma. Thank you! Just want to check my answer, haven't done sequences since high school. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #655738 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=655738\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "This is a geometric series of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ar%5En ALIGN=MIDDLE  ALT=\"a%2Ar%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > where n is a whole number. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F3 ALIGN=MIDDLE  ALT=\"a+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-4%2F3 ALIGN=MIDDLE  ALT=\"r+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The rule is this: an infinite geometric series converges to a fixed number if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C+r+%3C+1 ALIGN=MIDDLE  ALT=\"-1+%3C+r+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+-4%2F3+=+-1.33 ALIGN=MIDDLE  ALT=\"r+=+-4%2F3+=+-1.33\"   DISABLED_style= \"max-width:90%; height:auto;\" > (approx) is NOT in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+%3C+r+%3C+1 ALIGN=MIDDLE  ALT=\"-1+%3C+r+%3C+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the infinite geometric series does NOT converge.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the geometric series <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sum_{k=1}^{\infty}\frac{1}{3}\left(-\frac{4}{3}\right)^k\"> <font color=red size=4>diverges</font>. It will not approach a fixed number><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );