title: How can one obtain more accurate numerical approximations to an integral involving the floor function in Wolfram Alpha?
body: <p>After reading sections of the following <a href="https://xn--webducation-dbb.com/wp-content/uploads/2020/01/Problem-Books-in-Mathematics-Ovidiu-Furdui-auth.-Limits-Series-and-Fractional-Part-Integrals_-Problems-in-Mathematical-Analysis-Springer-Verlag-New-York-2013.pdf" rel="nofollow noreferrer">book</a> by Furdui and a <a href="https://artofproblemsolving.com/community/c7h1140833p5357644" rel="nofollow noreferrer">page</a> on the AoPS forum, I got interested in the integral</p>
  <p>                                                                  <a href="https://i.stack.imgur.com/xt4S7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xt4S7.png" alt="enter image description here" /></a></p>
  <p>Let's denote the value of this integral by A. Then we have that the value to 20 digits approximately comes down to:</p>
  <blockquote>
  <p>A = 0.6449340668482264365</p>
  </blockquote>
  <p>However, when one puts in the code</p>
  <pre><code>integrate 1/(floor(1/x)) from x=0 to 1 to 20 digits
  </code></pre>
  <p>In Wolfram Alpha, it generates the following approximation:</p>
  <blockquote>
  <p>A* = 0.64493701331278272222</p>
  </blockquote>
  <p>As one can see, the numbers start to deviate from one another after the seventh digit. So my question is:</p>
  <blockquote>
  <p><strong>Question</strong>: how, if at all, can one obtain more accurate numerical approximations to the value of the integral above in Wolfram Alpha?</p>
  </blockquote>