Find an expression in terms of n for summation

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SUMMARY

The discussion centers on finding an expression for the summation of \( \sum_{r=2}^{n} \frac{1}{r^2 - 1} \) using partial fractions. The user has successfully solved the first part but is seeking assistance with demonstrating that \( \sum_{r=1}^{n} \frac{1}{r^2} < \frac{7}{4} \) for all values of \( n \). The key to solving the second part lies in the results obtained from the first part of the summation.

PREREQUISITES
  • Understanding of partial fractions decomposition
  • Knowledge of summation notation and series
  • Familiarity with convergence of series
  • Basic calculus concepts related to inequalities
NEXT STEPS
  • Review partial fractions and their applications in summation
  • Study the convergence properties of series, particularly \( \sum_{r=1}^{n} \frac{1}{r^2} \)
  • Explore techniques for proving inequalities in series
  • Investigate the relationship between series and their bounds
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Students studying calculus, particularly those focusing on series and inequalities, as well as educators looking for examples of summation techniques and proofs.

elitewarr
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Homework Statement


By first expressing the general term in partial fractions, find an expression in terms of n for
summation of r=2 to n ( 1 / (r^2 - 1) ). Hence show that summation of r=1 to n ( 1 / r^2) i less than 7/4 for all values of n


Homework Equations





The Attempt at a Solution


I solved the first part of the question already. However, I'm stuck at the second part. Can anyone give me some clues to showing 1/r^2 sum from r = 1 to n is always less than 7/4?
Thank you.
 
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Well, what did you get for the first part? I suspect that is important to solving the second part!
 

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