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Limit $$\lim_{n \to \infty } \int_0 ^1(1-x)(1-x^2)\cdots(1-x^n)d x.$$

I tried to use the Beta function to calculate this limit, and I failed because I didn’t know how to calculate the gamma function like “gamma(1/n)”.

Similar method: how to compute this limit

I don’t know if it converges. But if it converges, I guess the limit is the 1/e. Because I try to use desmos to calculate the limited term and get a result like 1/e.

desmos pic

Feng
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Konan
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    This? https://math.stackexchange.com/q/1883140/42969, https://math.stackexchange.com/q/3019758/42969 – Martin R Sep 05 '23 at 09:23
  • According to the linked threads, the integral is $\frac{4\pi\sqrt3}{\sqrt{23}}\frac{\sinh\frac{\pi\sqrt{23}}3}{\cosh\frac{\pi\sqrt{23}}2} \approx 0.3684125359314337$, which is close, but not equal, to $1/e \approx 0.36787944117144233$. – Martin R Sep 05 '23 at 09:31
  • Oh yes, thank you Martin R, maybe I use different form and didn’t find the same problem. – Konan Sep 05 '23 at 09:35

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