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After reading sections of the following book by Furdui and a page on the AoPS forum, I got interested in the integral

                                                                  enter image description here

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?

Max Muller
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  • How do you know that the first number is more correct than the one given by Wolfram? – Nigel Jun 22 '22 at 20:16
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    @Nigel Because these digits match pi^(2)/6 - 1 for the first twenty digits - which is the closed form solution to the integral – Max Muller Jun 22 '22 at 20:28
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    May I suggest you use one free mathematica notebook in the cloud https://www.wolframcloud.com/?source=nav and something like: `NumberForm[NIntegrate[1/Floor[1/x], {x, 0, 1}, MaxRecursion -> 1000,WorkingPrecision -> 50, Method->"LocalAdaptive"] ,20]`. You might need to play with the precision and recursion parameters to avoid ending your free trial time. – Rodrigo Zepeda Jun 22 '22 at 21:19

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