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Let S denotes the sum of an infinite geo...

Let `S` denotes the sum of an infinite geometric progression whose first term is the value of the functions `f(x)=(sin(x-pi//6))/(sqrt3-2cosx)` at `x=pi/6,` if `f(x)` is continuous at `x=pi/6` and whose common ratio is the limiting value of the function `g(x)=(sin(x)^(1//3) ln(1+3x))/((arc tansqrtx)^2 (e^(5*x^(1//3))-1))` as `x rarr 0.` Find the value of `2S.`

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