A Weird Sum of Fractions!

Algebra Level 4

$\large{ S = \sum_{i=1}^n \dfrac{1}{(a_i)(a_i + a_{i+1})(a_i + a_{i+1} + a_{i+2}) \dotsm (a_i + a_{i+1} + \ldots + a_{i+n-2}) }}$

If $$a_1, a_2, \ldots, a_n$$ are real numbers with $$\sum_{i=1}^n a_i = 0$$, and where $$a_{n+1} = a_1, a_{n+2} = a_2$$ and so on.. assuming that the denominators are non-zero. Then find the value of $$S$$ upto three correct places of decimal.

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