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Find the sum of an infinite geometric se...

Find the sum of an infinite geometric series whose first term is `lim_(x->0) sum_(k=1)^(2011) {(x/(tan x)+2k)}/2011` and whose common ratio is the value of `lim_(x->0) (e^(tan^3x) - e^(x^3))/(2ln(1+x^3 sin^2 x))` [Note:(y) denotes fractional part of y].

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