Convergence Condition for Applying Ratio Test to Power Series

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SUMMARY

The ratio test can always be applied to a power series to determine convergence, specifically when evaluating the series of absolute values of its coefficients. This method proves absolute convergence by demonstrating that the series is bounded above and below by the series of absolute values and its negative. The discussion emphasizes the importance of applying the ratio test correctly to ensure accurate results in convergence analysis.

PREREQUISITES
  • Understanding of power series and their coefficients
  • Knowledge of the ratio test in calculus
  • Familiarity with absolute convergence concepts
  • Basic skills in mathematical analysis
NEXT STEPS
  • Study the application of the ratio test in various power series scenarios
  • Explore the concept of absolute convergence in more depth
  • Learn about alternative convergence tests for power series
  • Review examples of power series and their convergence properties
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and series convergence, as well as anyone seeking to deepen their understanding of power series analysis.

alexmahone
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Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
 
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Alexmahone said:
Given a power series, what is the condition on its coefficients that means the ratio test can be applied?

I think it can always be applied. Other test just may be easier for a given power series.
 
Alexmahone said:
Given a power series, what is the condition on its coefficients that means the ratio test can be applied?

You actually should always do the ratio test on the series of ABSOLUTE VALUES, to prove absolute convergence (i.e. to show the series is bounded both above and below by the series of absolute values and the negative of the series of absolute values).
 

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