SOLUTION: Determine if the limit exists as n goes to infinity: {{{ 1/(n+1)+1/(n+2)}}} +...+ {{{1/(2n-1) + 1/(2n)}}}. If the limit exists then find its value.

Question 1103369: Determine if the limit exists as n goes to infinity:
+...+ .
If the limit exists then find its value.
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
Let + � +
Consider + � +
For :

: :

That's n terms, each of which is less than 1.

which implies

For the lower bound: (all but the last term are > n/2)

So and the series has a limit as �> .
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To find the limit value, I cheated and used a one-line Perl script:
perl -e '$m=10000000; for($i=$m, $s=0; $i<(2*$m); $i++) { $s += 1/($i+1); } print "$s\n";'
0.693147155559907
perl -e '$m=100000000; for($i=$m, $s=0; $i<(2*$m); $i++) { $s += 1/($i+1); } print "$s\n";'
0.693147178059741
Which looks like ln(2).
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EDIT: I also should have shown that the series terms get smaller as �>
which goes to 0 monotonically as �> . This means increases by smaller amounts as n increases, and the series converges.
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