Series Laws and Theorems and how they affect each other

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SUMMARY

The discussion centers on the relationship between the Test for Divergence and the Integral Test in determining the convergence or divergence of series. It is established that a series can be initially classified as "possibly not divergent" using the Test for Divergence and later confirmed as divergent through the Integral Test. Specifically, the harmonic series 1/n exemplifies this, as it does not diverge according to the Test for Divergence but is indeed divergent. Thus, the Test for Divergence yields conclusions of either "definitely divergent" or "inconclusive."

PREREQUISITES
  • Understanding of series convergence and divergence
  • Familiarity with the Test for Divergence
  • Knowledge of the Integral Test for series
  • Basic concepts of mathematical analysis
NEXT STEPS
  • Study the properties of the harmonic series and its divergence
  • Learn about the application of the Integral Test in detail
  • Explore other convergence tests such as the Ratio Test and Comparison Test
  • Investigate the implications of series convergence in mathematical analysis
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced calculus or series analysis will benefit from this discussion.

kylera
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Is it mathematically possible for a series to be first declared "possibly not divergent" via the Test For Divergence, then to be declared divergent by applying the Integral Test?

On that tangent, does that mean that the Test For Divergence will only render the conclusions "definitely divergent" and "inconclusive"? Much thanks in advance.
 
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Yes to both questions.

For example, the harmonic series 1/n doesn't diverge via the Test for Divergence, but it is divergent.
 

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