document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1092428>Question 1092428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I determine whether the sequence converge or diverges. If it converges, give the limit. <BR>\n" );
document.write( "U1=1, U n+1= Un/3 for n&#8805;1 (1,n+1, n after U should be smaller size).<BR>\n" );
document.write( "Please help. Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #707086 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=707086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "if abs(r) < 1, then the geometric sum will converge.\r<BR>\n" );
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document.write( "if abs(r) > 1, then the geometric sum will not converge.\r<BR>\n" );
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document.write( "abs(r) < 1 means that:\r<BR>\n" );
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document.write( "r < 1 or -r < 1\r<BR>\n" );
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document.write( "solve for r in each of these equations and you get:\r<BR>\n" );
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document.write( "r < 1 is already solved for r.\r<BR>\n" );
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document.write( "start with -r < 1\r<BR>\n" );
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document.write( "multiply both sides of this equation by -1 to get:\r<BR>\n" );
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document.write( "r > -1\r<BR>\n" );
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document.write( "you get abs(r) < 1 when r < 1 and when r > -1.\r<BR>\n" );
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document.write( "this means -1 < r < 1\r<BR>\n" );
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document.write( "the geometric series will converge when that occurs.\r<BR>\n" );
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document.write( "since your common ratio is 1/3 which is > -1 and < 1, then your geometric series will converge.\r<BR>\n" );
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document.write( "the formula for the sum of a geometric series is:\r<BR>\n" );
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document.write( "Sn = A1 * (1 - r^n) / (1-r)\r<BR>\n" );
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document.write( "when -1 < r < 1, this series will converge to a limit as n approaches infinity.\r<BR>\n" );
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document.write( "that's because r^n will get closer and closer to 0 as n approaches infinity.\r<BR>\n" );
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document.write( "the limit of Sn as n approaches infinity when -1 < r < 1 is:\r<BR>\n" );
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document.write( "limit of Sn as n approaches infinity = 1 / (1-r).\r<BR>\n" );
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document.write( "since r = 1/3 in your problem, you get:\r<BR>\n" );
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document.write( "limit of Sn as n approaches infinity = 1 / (1-(1/3)).\r<BR>\n" );
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document.write( "this results in limit of Sn as n approaches infinity = 3/2 or 1.5.\r<BR>\n" );
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document.write( "here's a reference on the sum of a geometric series.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://mathematics.laerd.com/maths/geometric-progression-intro.php\" target = \"_blank\">http://mathematics.laerd.com/maths/geometric-progression-intro.php</a><BR>\n" );
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