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Incorrect result for improper integral with MMA?


The command:

Integrate[1/(x*Log[x]^2),{x,3,Infinity}]

yields the result:

\!\(\*
  RowBox[{
  \(Integrate::"idiv"\), \( : \ \), 
    "\<"Integral of \!\(1\/\(x\\ \(Log[x]\)\^2\)\) does not converge on \
\!\({3, \*InterpretationBox[\"\\[Infinity]\", DirectedInfinity[1]]}\)."\>"}]\)

i.e.  the integral does not converge.  However, analytic techniques seem
to indicate that the integral does indeed converge to 1/ln(3).  Mimicing
the analytic procedures step by step with mma also yields 1/ln(3).  Am I
being stupid, or is mma wrong?

E-mail responses gladly accepted.


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