For purposes of taking the floor (integer part) function,
n <= 1/y < n+1
1/(n+1) < y <= 1/n
The width (taking y as being laid out horizontally, as the independent variable) of a given rectangle making up part of the area will be
1/n - 1/(n+1) = 1/(n(n+1))
where the height is 1/n.
This makes the area of that portion of the integral equal to
a = 1/(n^2 * (n+1))
And the total area is the sum of these for n=1 to infinity.
Summing this via computer program
FOR n = 1 TO 1000000
t = t + 1 / (n * n * (n + 1))
PRINT n, t
IF t = prevT THEN
ctDown = ctDown + 1
IF ctDown > 10 THEN EXIT FOR
prevT = t
gives a numeric answer of .6449340668423813 as the approximation.
This was reached at n = 262,102 and remained constant, to that degree of accuracy, thereafter.
Posted by Charlie
on 2008-10-15 12:28:38