D'Alembert Ratio Test: Convergence test

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SUMMARY

The D'Alembert Ratio Test is utilized to determine the convergence of series, specifically applicable to sequences of positive numbers. The test requires the use of the modulus (absolute value) when calculating limits, ensuring that the results are valid for convergence analysis. A negative result from the test indicates divergence, as the ratio test is fundamentally designed to assess positive sequences. Thus, the application of the modulus is essential for accurate results in convergence testing.

PREREQUISITES
  • Understanding of the D'Alembert Ratio Test
  • Familiarity with limits in calculus
  • Knowledge of sequences and series
  • Concept of absolute values in mathematics
NEXT STEPS
  • Study the application of the D'Alembert Ratio Test in various series
  • Learn about convergence tests beyond the ratio test, such as the Root Test
  • Explore the implications of absolute convergence versus conditional convergence
  • Review examples of sequences that demonstrate the use of the modulus in convergence tests
USEFUL FOR

Mathematics students, educators, and anyone studying series convergence in calculus will benefit from this discussion.

ruby_duby
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Hi this is just a general question about using the ratio test for convergence.

If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative?

Or does the formula for the ratio test when finding the limit, use the modulus?
 
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Strictly speaking, the ratio test only applies to sequences of positive numbers so, yes, it applies to the modulus (absolute value).
 

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