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I am interested in finding $$f(n) = \int_0^1\int_0^1\ldots\int_0^1\frac{1}{1-\prod_{i=1}^{n}\ln(x_i)}\,dx_1dx_2\ldots dx_n$$ for positive integer $n$.

For example, $$f(2)=\int_0^1\int_0^1\frac{1}{1-\ln(x_1)\ln(x_2)}\,dx_1dx_2$$

I found $f(1) = e \cdot E_1(1) \approx 0.596$, $f(2)$ diverges, $f(3) \approx 0.724$. $$$$ My questions

How can I find $f(n)$ for positive integer $n$?

If that is not possible, what is an approximation?

I know for sure that if $n$ is odd, $f(n)$ converges. This is because $\frac{1}{1-\prod_{i=1}^{n}\ln(x_i)}$ will converge for $0 \le x_i \le 1$. I also suspect that if $n$ is even, $f(n)$ diverges.

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  • $\begingroup$ Can you give some background about this integral? Like how did you encounter it and why is it in your interested? Because it doesn't look easy. $\endgroup$ – Zacky Aug 7 '19 at 8:34
  • $\begingroup$ @ㄴㄱ It just popped into my head after seeing another question that was similar. Unfortunately, I don't remember what that question was. $\endgroup$ – automaticallyGenerated Aug 7 '19 at 17:15

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