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  3. example 4 determine whether the series below converges or diverges...

Question: example 4 determine whether the series below converges or diverges...

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EXAMPLE 4 Determine whether the series below converges or diverges oo In(2n) n-1 5n SOLUTION The function f(x) In(2x)/(5x) is positive and continuous for x > function is continuous. But it is not obvious whether f is decreasing, so we compute its derivativee because the logarithm 5x 5ln(2x) f (x) = Thus f (x) < 0 when In(2x) > , that is, x > . It follows that fis decreasing when x > and so we can apply the Integral Test. In (2 ) In (2 -dr= lim = lim 10 (In (2 t))In(2) lim ---oo 10 10 Since this improper integral is divergent, the series is also divergent by the integral test.

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