Question: EXAMPLE 4 Determine whether the series n=1ln(3n)2n converges or diverges.SOLUTION The function f(x)=ln(3x)(2x)is positive and continuous for x>because the logarithm function iscontinuous. But itis not

EXAMPLE 4 Determine whether the series n=1ln(3n)2n converges or diverges.SOLUTION The function f(x)=ln(3x)(2x)is positive and continuous for x>because the logarithm function iscontinuous. But itis not obvious whether or not fis decreasing, sowe compute its derivative:Thus f'(x)<0 when ln(3x)>,, that is,x>.It follows that fis decreasing whenx>and sowe can apply the Integral Test:1ln(3x)2xdx=limt1tln(3x)2xdx=limt|1t4|1=limt(ln(3t))24-(ln(3))24=Since this improper integral is divergent, the series n=1ln(3n)2nis also divergent by the Integral Test.

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