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Question 1: Box answers please

EXAMPLE 4 Determine whether the series In(3n) converges or diverges. n=1 3n SOLUTION The function f(x) = In(3x) (3x) is positive and continuous for x > because the logarithm function is continuous. But it is not obvious whether or not f is decreasing, so we compute its derivative: f' ( x ) = 3x - 3 In ( 3x ) 9x 3x2 Thus f'(x) that is, x > It follows that f is decreasing when x > and so we can apply the Integral Test: In (3X) dx = lim it In(3x) dx 3x t-+ 00 1 3x = lim t - 00 6 = lim (In(3t))2 _ (In(3) )