Whoa! That's Insane!

Calculus Level 3

An=01010101 ⁣01i=1nai1i=1naida1da2da3dan, A_n= \int_0^1 \int_0^1\int_0^1\int_0^1\dotsi \int_0^1 {\frac{\prod_{i=1}^{n} a_i}{1-\prod_{i=1}^{n} a_i}} da_1da_2da_3\dotsc da_n, where the integral sign is repeated nn times.

Find n=2An\displaystyle \sum_{n=2}^{\infty} A_n.

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