What is the name for testing divergence in rational functions?

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Discussion Overview

The discussion revolves around identifying a specific name for a method of testing divergence in rational functions, particularly in the context of infinite series. Participants explore the relationship between the degrees of the numerator and denominator in determining convergence or divergence.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that if the difference in the degree of the functions (bottom - top) is less than or equal to 1, then the series is divergent, while if the difference is greater than 1, it is convergent.
  • Another participant interprets this as a description of the ratio test.
  • A different participant clarifies that while it is similar to the ratio test, it provides a stronger assertion about divergence based on the degree difference.
  • One participant asserts that the method described is actually the comparison test, not the ratio test.
  • Another participant reiterates that the method is the comparison test, followed by mentioning the p-series test.

Areas of Agreement / Disagreement

Participants express disagreement regarding the classification of the method discussed, with some asserting it is the comparison test and others suggesting it relates to the ratio test. The discussion remains unresolved regarding the correct terminology.

Contextual Notes

There are limitations in the definitions and assumptions regarding the tests being discussed, as well as the specific conditions under which the divergence or convergence is determined.

erich76
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After doing my homework on testing for convergence/divergence in infinite series, I noticed that if you are testing for divergence of a rational function, if the difference in the degree of the functions (bottom - top) \leq 1, then it is divergent, and if the difference in the degree of the functions is > 1, then it is convergent. I can post this proof if someone wants.

I was just wondering if there is a name for this concept..
 
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From your question (as best as I can understand it), it seems to be a description of the ratio test.
 
It's similar to the ratio test, but all the ratio test says is that if the limit comes out to a positive finite number, then both of the series either diverge, or both converge. I used the ratio test in proving this, but this goes a step further and says that if the difference in degrees is less than or equal to 1, it diverges no matter what. And if it is greater than 1, it converges no matter what.
 
No, what you give is the "comparison test", not the "ratio test".
 
HallsofIvy said:
No, what you give is the "comparison test", not the "ratio test".

Followed by the p-series test.
 

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