Integral for Multiple Natural Logs

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SUMMARY

The discussion focuses on determining the convergence or divergence of the series Ʃ1/(n(ln n)(ln(ln(n)))) from 2 to infinity using the Integral Test. The user is struggling with integration by parts for the function 1/(ln x)ln(ln(x)). A suggested substitution is u = ln(ln(x)), leading to du = (1/ln(x))(1/x)dx, which simplifies the integral for further analysis.

PREREQUISITES
  • Understanding of the Integral Test for convergence
  • Familiarity with integration by parts
  • Knowledge of natural logarithms and their properties
  • Basic calculus concepts, including substitution methods
NEXT STEPS
  • Research the Integral Test for series convergence
  • Practice integration by parts with logarithmic functions
  • Explore advanced techniques for simplifying integrals involving logarithms
  • Learn about convergence tests for series beyond the Integral Test
USEFUL FOR

Students studying calculus, particularly those focusing on series convergence, and anyone seeking to improve their skills in integration techniques involving logarithmic functions.

Dragon M.
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Homework Statement



I need to find whether or not the Ʃ1/(n(ln n)(ln(ln(n)))) (from 2 to infinity) converges or diverges using the Integral test. I think the best question to ask is where to start with the integral?


Homework Equations



None.


The Attempt at a Solution



I've tried integrating by parts, but I can't find du or v for 1/(lnx)ln(ln(x)). Is there a way to simplify this into a more friendly form?
 
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If you let u= ln(ln(x)) then [itex]du= (1/ln(x))(1/x)dx[/itex].
 

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