document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205331>Question 1205331</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of the following series:\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842057 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=842057\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Each term is of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28sqrt%28k%29%2Bsqrt%28k%2B1%29%29 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%28k%29%2Bsqrt%28k%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k ranges from k = 1 to k = 35.\r<BR>\n" );
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document.write( "Rationalize the denominator, by multiplying top and bottom by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28k%29-sqrt%28k%2B1%29 ALIGN=MIDDLE  ALT=\"sqrt%28k%29-sqrt%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and you should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28k%2B1%29-sqrt%28k%29 ALIGN=MIDDLE  ALT=\"sqrt%28k%2B1%29-sqrt%28k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > after the simplification is done. <BR>\n" );
document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28sqrt%28k%29%2Bsqrt%28k%2B1%29%29+=+sqrt%28k%2B1%29-sqrt%28k%29 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%28k%29%2Bsqrt%28k%2B1%29%29+=+sqrt%28k%2B1%29-sqrt%28k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll leave the scratch work for the student to do.\r<BR>\n" );
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document.write( "If k = 1, then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%281%2B1%29-sqrt%281%29=+sqrt%282%29-sqrt%281%29 ALIGN=MIDDLE  ALT=\"sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%281%2B1%29-sqrt%281%29=+sqrt%282%29-sqrt%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If k = 2, then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%282%2B1%29-sqrt%282%29=+sqrt%283%29-sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%282%2B1%29-sqrt%282%29=+sqrt%283%29-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If k = 3, then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%283%2B1%29-sqrt%283%29=+sqrt%284%29-sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%28k%2B1%29-sqrt%28k%29+=+sqrt%283%2B1%29-sqrt%283%29=+sqrt%284%29-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding those terms gets us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%282%29-sqrt%281%29%29+%2B+%28sqrt%283%29-sqrt%282%29%29%2B%28sqrt%284%29-sqrt%283%29%29+=+sqrt%283%29-sqrt%281%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The inner terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancel out.<BR>\n" );
document.write( "This will extend to the summation of terms k = 1 through k = 35<BR>\n" );
document.write( "The inner terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ... , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2834%29 ALIGN=MIDDLE  ALT=\"sqrt%2834%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2835%29 ALIGN=MIDDLE  ALT=\"sqrt%2835%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancel leaving behind <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%29-sqrt%281%29+=+6-1+=+5 ALIGN=MIDDLE  ALT=\"sqrt%2836%29-sqrt%281%29+=+6-1+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For more info and examples, search out \"telescoping series\".<BR>\n" );
document.write( "The name is due to the fact the long series collapses to a very small expression since the inner terms cancel out.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://mathworld.wolfram.com/TelescopingSum.html\">https://mathworld.wolfram.com/TelescopingSum.html</a>\r<BR>\n" );
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document.write( "Answer: 5<BR>\n" );
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