document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=196526>Question 196526</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem.<BR>\n" );
document.write( "It says, Decide whether each infinite geometric series diverges or converges. State whether each series has a sum.\r<BR>\n" );
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document.write( "I understand that it diverges when the absolute value of r is > or = 1 and converges when the absolute value of r is < 1, but how do you know whether each series has a sum. For example one problem says:<BR>\n" );
document.write( "1. 4 + 2 + 1 + ...<BR>\n" );
document.write( "and I know that r=2 therefore it diverges but I can not figure out whether it has a sum or not.<BR>\n" );
document.write( "Thanks for your time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147322 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=147322\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well first off, if a series diverges, it does not have a sum (ie the if you add up the infinite terms, you won't get a fixed number).\r<BR>\n" );
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document.write( "In this problem, however, the series does converge. Here's why:\r<BR>\n" );
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document.write( "The sequence of terms: 4, 2, 1, ... can be modeled by the formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=4%2A%281%2F2%29%5En ALIGN=MIDDLE  ALT=\"a%5Bn%5D=4%2A%281%2F2%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"n\" starts at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: plug in some values to test this)\r<BR>\n" );
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document.write( "Since this equation fits the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%2Ar%5En ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%2Ar%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2F2%3C1 ALIGN=MIDDLE  ALT=\"r=1%2F2%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the absolute value of \"r\" is less than 1. So the series converges.\r<BR>\n" );
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document.write( "To find the sum of the infinite series, simply use the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=a%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S=a%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where \"a\" is the first term and \"r\" is given as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the infinite sum is...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=4%2F%281-1%2F2%29=4%2F%281%2F2%29=4%282%29=8 ALIGN=MIDDLE  ALT=\"S=4%2F%281-1%2F2%29=4%2F%281%2F2%29=4%282%29=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words, \r<BR>\n" );
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document.write( "4+2+1+...=8\r<BR>\n" );
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document.write( "I hope this makes sense. </SPAN>\n" );
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