document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=29292>Question 29292</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that 1/sqrt 1 + 1/sqrt 2 + 1/sqrt 3...+ 1/sqrt n < 2*sqrt n for all positive integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16146 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16146\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Show that 1/sqrt 1 + 1/sqrt 2 + 1/sqrt 3...+ 1/sqrt n < 2*sqrt n for all positive integers.<BR>\n" );
document.write( "COSIDER THE EQN.<BR>\n" );
document.write( "SQRT(N+1)-SQRT(N)={SQRT(N+1)-SQRT(N)}*{SQRT(N+1)+SQRT(N)/{SQRT(N+1)+SQRT(N)}<BR>\n" );
document.write( "={(N+1)-(N)}/{SQRT(N+1)+SQRT(N)}=1 / {SQRT(N+1)+SQRT(N)}>1 / {SQRT(N+1)+SQRT(N+1)}<BR>\n" );
document.write( "=1 / 2*SQRT(N+1)...HENCE<BR>\n" );
document.write( "SQRT(N+1)-SQRT(N) > 1/2*SQRT(N+1)..PUT N=1,2,3...N..IN THIS EQN. AND ADD UP...<BR>\n" );
document.write( "N=1...............SQRT2-SQRT1>1/2SQRT2<BR>\n" );
document.write( "N=2...............SQRT3-SQRT2>1/2SQRT3<BR>\n" );
document.write( "N=3...............SQRT4-SQRT3>1/2SQRT4<BR>\n" );
document.write( "......................................<BR>\n" );
document.write( ".......................................<BR>\n" );
document.write( "N=N-1.............SQRT(N)-SQRT(N-1)>1/2SQRT(N)<BR>\n" );
document.write( "N=N...............SQRT(N+1)-SQRT(N)>1/2SQRT(N+1)<BR>\n" );
document.write( "-------------------------------------------------ADDING.....<BR>\n" );
document.write( "WE FIND ALL TERMS ON LHS CANCEL EXCEPT	<BR>\n" );
document.write( "SQRT(N+1)-SQRT(1)>(1/2){ 1/sqrt 2 + 1/sqrt 3...+ 1/sqrt n }	<BR>\n" );
document.write( "OR....	\r<BR>\n" );
document.write( "\n" );
document.write( "{1/sqrt 1 + 1/sqrt 2 + 1/sqrt 3...+ 1/sqrt n } > 2*{SQRT(N+1)-SQRT(1)}+1/SQRT1<BR>\n" );
document.write( "=2SQRT(N+1)-1>2SQRT(N+1)>2SQRT(N)	\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );