Solving the Harmonic Series and Ratio Problem

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SUMMARY

The discussion focuses on solving the harmonic series and ratio problem represented by the infinite series from n=4 of the expression (2/n) - (2/(n-1)). Participants conclude that the series converges to -2/3. They suggest using the concept of a telescoping series to simplify the problem and recommend applying the ratio test to analyze convergence. The initial terms of the series help illustrate the behavior as n approaches infinity.

PREREQUISITES
  • Understanding of harmonic series
  • Familiarity with telescoping series
  • Knowledge of the ratio test for convergence
  • Basic calculus concepts, particularly limits
NEXT STEPS
  • Study the properties of harmonic series in detail
  • Learn about telescoping series and their applications
  • Review the ratio test for series convergence
  • Practice solving similar infinite series problems
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in series convergence and analysis techniques.

Kyousik
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Homework Statement


(infinity)sigma(n=4) [ (2 / n) - (2 / (n -1)) ]


Homework Equations


Harmonic series and ratio?


The Attempt at a Solution


It's supposed to converge to -2/3, but, I don't know how.

The first compare to a harmonic series and we see that goes to 0.

The second, we use a ratio test or something and then compare it? If we simply take the limit it goes to 0.

I don't know how to show it.
 
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Write out the first few terms of the series and see if anything occurs to you.
 
"Telescoping series"
 

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