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Summing inverses (Posted on 2004-08-19) Difficulty: 3 of 5
What's the limit, as n→∞, of 1/(n+1)+1/(n+2)+1/(n+3)+...+1/(2n)?

See The Solution Submitted by Federico Kereki    
Rating: 4.0000 (5 votes)

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Solution Puzzle Solution | Comment 17 of 18 |
(In reply to Answer by K Sengupta)

We know that the nth harmonic number is given asymptotically by:

H(n) ~ ln n + gamma + (2n)^-1 - 1/12*(n^-2) + (n^-4)/120 - (n^-6)/252 +  ......, where gamma represents Euler-Mascheroni constant

(Reference:
http://mathworld.wolfram.com/HarmonicNumber.html )

Then, it follows that:

H(2n) - H(n) 
-> ln (2n) - ln(n) = ln 2, as n -> infinity

or, 1/(n+1)+1/(n+2)+1/(n+3)+...+1/(2n) -> ln 2 as n ->  infinity

Edited on August 28, 2008, 1:32 pm
  Posted by K Sengupta on 2008-08-28 13:00:28

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