document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=42555>Question 42555</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of the infinite geometric series: 1 + 3/5 + 9/25 + ..., if it exists.\r<BR>\n" );
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document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27669 by </B><A HREF=/tutors/aboutme.mpl?userid=psbhowmick><B>psbhowmick(878)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=psbhowmick><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=psbhowmick><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=27669\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sum of the first 'n'-terms of a geometric series: a, ar, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2 ALIGN=MIDDLE  ALT=\"ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E3 ALIGN=MIDDLE  ALT=\"ar%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,.........,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"ar%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+a%2A%281-r%5En%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S+=+a%2A%281-r%5En%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%280+%3C+r+%3C+1%29 ALIGN=MIDDLE  ALT=\"red%280+%3C+r+%3C+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "When 'n' is very large, <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;\" > << 1 [means <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;\" > is very very less than 1].<BR>\n" );
document.write( "So, obviously when 'n' is infinity, <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;\" > << 1 so it can be neglected (means taken as 0) in comparison to 1.<BR>\n" );
document.write( "Thus the summation formula for 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;\" > becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+a%2A%281-0%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S+=+a%2A%281-0%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <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;\" > \r<BR>\n" );
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document.write( "Here, a = 1 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28r=3%2F5%29 ALIGN=MIDDLE  ALT=\"red%28r=3%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%280+%3C+r+%3C+1%29 ALIGN=MIDDLE  ALT=\"red%280+%3C+r+%3C+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Hence the formula of summation of geometric series for infinite number of terms (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;\" >) is applicable.\r<BR>\n" );
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document.write( "Thus the summation of the given infinite geometric series is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+1%2F%281-3%2F5%29 ALIGN=MIDDLE  ALT=\"S+=+1%2F%281-3%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+1%2F%282%2F5%29 ALIGN=MIDDLE  ALT=\"S+=+1%2F%282%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+5%2F2+=+2.5 ALIGN=MIDDLE  ALT=\"S+=+5%2F2+=+2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, the summation of the given series exists and its value is 2.5. </SPAN>\n" );
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