Inverse Trigonometric Series Product!

Geometry Level 5

\[\large{ S_n = \prod_{k=1}^n \ \dfrac{\arcsin \left( \frac{9k+2}{\sqrt{27k^3 + 54k^2 + 36k + 8}} \right) }{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)} }\]

Find the value of \(\large{ \ln(S_{2015}) }\) correct upto three places of decimal.

Bonus: Generalize for \(S_n, n \in \mathbb Z^+\).

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