document.write( "Question 1103369: Determine if the limit exists as n goes to infinity:\r
\n" ); document.write( "\n" ); document.write( " $\"+1%2F%28n%2B1%29%2B1%2F%28n%2B2%29\"$ +...+ $\"1%2F%282n-1%29+%2B+1%2F%282n%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "If the limit exists then find its value. \n" ); document.write( "

Algebra.Com's Answer #718183 by math_helper(2461) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let $\"+S%5Bn%5D+=+1%2F%28n%2B1%29+%2B+1%2F%28n%2B2%29+\"$ + Е + $\"+1%2F%282n-1%29+%2B+1%2F2n+\"$ \r
\n" ); document.write( "\n" ); document.write( "Consider $\"+nS%5Bn%5D+=+n%2F%28n%2B1%29+%2B+n%2F%28n%2B2%29+\"$ + Е + $\"+n%2F%282n-1%29+%2B+n%2F2n+\"$
\n" ); document.write( "For $\"+n%3E=1+\"$ :
\n" ); document.write( " $\"+n%2F%28n%2B1%29+%3C+1+\"$
\n" ); document.write( " $\"+n%2F%28n%2B2%29+%3C+1+\"$
\n" ); document.write( ": :
\n" ); document.write( " $\"+n%2F%282n%29+%3C+1+\"$ \r
\n" ); document.write( "\n" ); document.write( "That's n terms, each of which is less than 1.
\n" ); document.write( " $\"+nS%5Bn%5D+%3C+n+\"$
\n" ); document.write( "which implies
\n" ); document.write( " $\"+S%5Bn%5D+%3C+1+\"$ \r
\n" ); document.write( "\n" ); document.write( "For the lower bound: $\"+nS%5Bn%5D+%3E=+n%2F2+\"$ (all but the last term are > n/2)
\n" ); document.write( " $\"+S%5Bn%5D+%3E+1%2F2+\"$ \r
\n" ); document.write( "\n" ); document.write( "So $\"+1%2F2+%3C+S%5Bn%5D+%3C+1+\"$ and the series has a limit as $\"+n+\"$ Ч> $\"+infinity+\"$ .
\n" ); document.write( "ЧЧЧЧЧЧЧЧЧЧЧ
\n" ); document.write( "To find the limit value, I cheated and used a one-line Perl script:
\n" ); document.write( "perl -e '$m=10000000; for($i=$m, $s=0; $i<(2*$m); $i++) { $s += 1/($i+1); } print \"$s\n\";'
\n" ); document.write( "0.693147155559907
\n" ); document.write( "perl -e '$m=100000000; for($i=$m, $s=0; $i<(2*$m); $i++) { $s += 1/($i+1); } print \"$s\n\";'
\n" ); document.write( "0.693147178059741\r
\n" ); document.write( "\n" ); document.write( "Which looks like ln(2). \r
\n" ); document.write( "\n" ); document.write( "ЧЧЧЧЧЧЧЧЧЧЧЧ
\n" ); document.write( "EDIT: I also should have shown that the series terms get smaller as $\"+n+\"$ Ч> $\"+infinity+\"$
\n" ); document.write( " $\"+S%5Bn%5D+-+S%5Bn-1%5D+=+1%2F%282n%2A%282n-1%29%29++\"$ which goes to 0 monotonically as $\"+n\"$ Ч> $\"+infinity+\"$ . This means $\"+S%5Bn%5D+\"$ increases by smaller amounts as n increases, and the series converges. \n" ); document.write( "

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