document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168321>Question 1168321</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If S=3/2+5/4+7/8+...+(2n+1)/2^n+... where n=1, 2, 3, ..., then find the value of S. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792925 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=792925\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Re-visit... \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5BN%5D+=+sum%28%282k%2B1%29%2F2%5Ek%2C+1%2C+N%29+ ALIGN=MIDDLE  ALT=\"S%5BN%5D+=+sum%28%282k%2B1%29%2F2%5Ek%2C+1%2C+N%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sum%28%28%282k%29%2F2%5Ek%29%2C1%2CN%29+%2B+sum%28%281%2F%282%5Ek%29%29%2C+1%2C+N%29 ALIGN=MIDDLE  ALT=\"+sum%28%28%282k%29%2F2%5Ek%29%2C1%2CN%29+%2B+sum%28%281%2F%282%5Ek%29%29%2C+1%2C+N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F2%5EN%29%28-N%2B2%5E%28N%2B1%29-2%29+%2B+%281-2%5E%28-N%29%29+ ALIGN=MIDDLE  ALT=\"+%282%2F2%5EN%29%28-N%2B2%5E%28N%2B1%29-2%29+%2B+%281-2%5E%28-N%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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document.write( "S = Lim(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;\" >) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%282%2F2%5EN%29%28-N%2B2%5E%28N%2B1%29-2%29+%2B+%281-2%5E%28-N%29%29%29+ ALIGN=MIDDLE  ALT=\"+%282%2F2%5EN%29%28-N%2B2%5E%28N%2B1%29-2%29+%2B+%281-2%5E%28-N%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "S = Lim(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;\" >) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28%28-2N%2F2%5EN%29%2B%282%5E%28N%2B2%29%2F2%5EN%29-%282%2F2%5EN%29%29+%2B+%281-2%5E%28-N%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "S = Lim(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;\" >) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28red%28-2N%2F2%5EN%29%2B%282%5E%28N%2B2%29%2F2%5EN%29-red%282%2F2%5EN%29%29+%2B+%281-red%282%5E%28-N%29%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<br><BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28red%29 ALIGN=MIDDLE  ALT=\"red%28red%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms go to zero,  leaving 4+1 = 5<BR>\n" );
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document.write( "I still needed help (WolframAlpha) to get the first partial sum.  \r<BR>\n" );
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document.write( "While I can't derive the answer, here is a little script that sums the first 1000 terms.  I verified this answer on Wolfram Alpha.<BR>\n" );
document.write( "<br>\r<BR>\n" );
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document.write( "perl -e '$n=1000;$s=0;for($i=1;$i<=$n;$i++) {$s+=(2*$i+1)/(2**$i); if ($i<=20 || !($i%100)) { print \"$i terms: sum is $s\n\";}} '<BR>\n" );
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document.write( "1 terms: sum is 1.5<BR>\n" );
document.write( "2 terms: sum is 2.75<BR>\n" );
document.write( "3 terms: sum is 3.625<BR>\n" );
document.write( "4 terms: sum is 4.1875<BR>\n" );
document.write( "5 terms: sum is 4.53125<BR>\n" );
document.write( "6 terms: sum is 4.734375<BR>\n" );
document.write( "7 terms: sum is 4.8515625<BR>\n" );
document.write( "8 terms: sum is 4.91796875<BR>\n" );
document.write( "9 terms: sum is 4.955078125<BR>\n" );
document.write( "10 terms: sum is 4.9755859375<BR>\n" );
document.write( "11 terms: sum is 4.98681640625<BR>\n" );
document.write( "12 terms: sum is 4.992919921875<BR>\n" );
document.write( "13 terms: sum is 4.9962158203125<BR>\n" );
document.write( "14 terms: sum is 4.99798583984375<BR>\n" );
document.write( "15 terms: sum is 4.99893188476562<BR>\n" );
document.write( "16 terms: sum is 4.99943542480469<BR>\n" );
document.write( "17 terms: sum is 4.99970245361328<BR>\n" );
document.write( "18 terms: sum is 4.99984359741211<BR>\n" );
document.write( "19 terms: sum is 4.99991798400879<BR>\n" );
document.write( "20 terms: sum is 4.99995708465576<BR>\n" );
document.write( "100 terms: sum is 5<BR>\n" );
document.write( "200 terms: sum is 5<BR>\n" );
document.write( "300 terms: sum is 5<BR>\n" );
document.write( "400 terms: sum is 5<BR>\n" );
document.write( "500 terms: sum is 5<BR>\n" );
document.write( "600 terms: sum is 5<BR>\n" );
document.write( "700 terms: sum is 5<BR>\n" );
document.write( "800 terms: sum is 5<BR>\n" );
document.write( "900 terms: sum is 5<BR>\n" );
document.write( "1000 terms: sum is 5 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );