After reading sections of the following book by Furdui and a page on the AoPS forum, I got interested in the integral
Let's denote the value of this integral by A. Then we have that the value to 20 digits approximately comes down to:
A = 0.6449340668482264365
However, when one puts in the code
integrate 1/(floor(1/x)) from x=0 to 1 to 20 digits
In Wolfram Alpha, it generates the following approximation:
A* = 0.64493701331278272222
As one can see, the numbers start to deviate from one another after the seventh digit. So my question is:
Question: how, if at all, can one obtain more accurate numerical approximations to the value of the integral above in Wolfram Alpha?