document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209795>Question 1209795</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of the series<BR>\n" );
document.write( "$$1 + \frac{1}{2} + \frac{1}{10} + \frac{1}{20} + \frac{1}{100} + \cdots,$$<BR>\n" );
document.write( "where we alternately multiply by $\frac 12$ and $\frac 15$ to get successive terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850493 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850493\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1%2F2%2B1%2F10%2B1%2F20%2B1%2F100%2B1%2F200 ALIGN=MIDDLE  ALT=\"1%2B1%2F2%2B1%2F10%2B1%2F20%2B1%2F100%2B1%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...<br><BR>\n" );
document.write( "Split the series into two purely geometric series:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1%2F10%2B1%2F100 ALIGN=MIDDLE  ALT=\"1%2B1%2F10%2B1%2F100\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...<br><BR>\n" );
document.write( "and<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%2B1%2F20%2B1%2F200 ALIGN=MIDDLE  ALT=\"1%2F2%2B1%2F20%2B1%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...<br><BR>\n" );
document.write( "The infinite sum of the first series is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-1%2F10%29=1%2F%289%2F10%29=10%2F9 ALIGN=MIDDLE  ALT=\"1%2F%281-1%2F10%29=1%2F%289%2F10%29=10%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note the second series is just half of the first, so the sum of the second series is 5/9.<br><BR>\n" );
document.write( "The sum of the original series is then 10/9 + 5/9 = 15/9 = 5/3.<br><BR>\n" );
document.write( "ANSWER: 5/3<br><BR>\n" );
document.write( "Alternatively, we could group the terms in pairs to obtain a single purely geometric series.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1%2F2=3%2F2 ALIGN=MIDDLE  ALT=\"1%2B1%2F2=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F10%2B1%2F20=3%2F20 ALIGN=MIDDLE  ALT=\"1%2F10%2B1%2F20=3%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F100%2B1%2F200%2B3%2F200 ALIGN=MIDDLE  ALT=\"1%2F100%2B1%2F200%2B3%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The given series is then equivalent to the series<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2%2B3%2F20%2B3%2F200 ALIGN=MIDDLE  ALT=\"3%2F2%2B3%2F20%2B3%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...<br><BR>\n" );
document.write( "The sum of that series is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29%2F%281-1%2F10%29=%283%2F2%29%2F%289%2F10%29=%283%2F2%29%2810%2F9%29=30%2F18=5%2F3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );