What's the limit, as n→∞, of 1/(n+1)+1/(n+2)+1/(n+3)+...+1/(2n)?

f(n) = f(n-1) + 1/(2n-1) +1/2n - 1/n
      = f(n-1) + 1/(2n-1) - 1/2n

f(1) = 1/2 = 1 - 1/2

Therefore, by induction,

f(n) = 1 - 1/2 + 1/3 - 1/4 ...  + 1/(2n-1) - 1/2n

The requested limit =
  1 - 1/2 + 1/3 - 1/4 ...  (for an infinite number of terms)

It is well known that
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 ...

If x = 1, then

ln(2) = 1 - 1/2 + 1/3 - 1/4 ... 
        = the requested limit

Posted by Steve Herman on 2004-10-14 22:28:25