document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141362>Question 1141362</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello! I’m really stumped on this. The question says to consider the infinite geometric series:\r<BR>\n" );
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document.write( "sigma infinity n=1 -4(1/3)^n-1\r<BR>\n" );
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document.write( "and it says the lower limit of the summation notation is “n=1”. It then says to write the first four terms of the series, asks if the series diverge or converge, and if the series has a sum, find it. I think the series converge, and if I solved for the sum correctly, I got -5.9..., but I can’t figure out the first four terms right. Can you help? And have I solved the rest correctly? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #761902 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=761902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assuming this is right:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sum%28-4%281%2F3%29%5E%28n-1%29%2C+n=1%2Cinfinity%29+ ALIGN=MIDDLE  ALT=\"+sum%28-4%281%2F3%29%5E%28n-1%29%2C+n=1%2Cinfinity%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "then your answer for the sum of the first four terms is correct (more precisely, the answer is -160/27 which is about -5.925925...).<br>\r<BR>\n" );
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document.write( "The sum of n terms can be found from this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5Bn%5D+=+a%5B1%5D%2A%281-r%5En%29%2F%281-r%29+ ALIGN=MIDDLE  ALT=\"+S%5Bn%5D+=+a%5B1%5D%2A%281-r%5En%29%2F%281-r%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <BR>\n" );
document.write( "where r is the common ratio (1/3) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term (-4) ( so you do need to plug in n=1 to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > )\r<BR>\n" );
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document.write( "Example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B2%5D+=+-4%281-%281%2F3%29%5E2%29%2F%281-1%2F3%29+=+-4%28%288%2F9%29%2F%282%2F3%29%29+=+-4%284%2F3%29+=+-16%2F3+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You could also plug in n=1,2,3,4 into the summation and write it all out long-hand but its a lot of extra work.<br>\r<BR>\n" );
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document.write( "As n--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5En ALIGN=MIDDLE  ALT=\"r%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > term goes to zero, leaving just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%2F%281-r%29+ ALIGN=MIDDLE  ALT=\"a%5B1%5D%2F%281-r%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the sum of the infinite progression:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sum%28-4%281%2F3%29%5E%28n-1%29%2C+n=1%2Cinfinity%29+=+%28a%5B1%5D%2F%281-r%29%29+=+%28-4%29+%2F+%281-%281%2F3%29%29+=+-12%2F2+=+highlight%28-6%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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