What is the name for testing divergence in rational functions?

  • Thread starter erich76
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    Divergence
In summary, after researching how to test for convergence or divergence in infinite series, it was discovered that for rational functions, if the difference in degrees between the top and bottom functions is less than or equal to 1, the series is divergent. If the difference is greater than 1, the series is convergent. This concept is similar to the ratio test, but takes it a step further by stating that if the difference in degrees is less than or equal to 1, the series will always diverge, and if it is greater than 1, it will always converge. This is known as the "comparison test", not the "ratio test". The conversation also touched on the p-series test.
  • #1
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) [itex]\leq[/itex] 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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  • #2
From your question (as best as I can understand it), it seems to be a description of the ratio test.
 
  • #3
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.
 
  • #4
No, what you give is the "comparison test", not the "ratio test".
 
  • #5
HallsofIvy said:
No, what you give is the "comparison test", not the "ratio test".

Followed by the p-series test.
 

1. What is divergence in science?

Divergence in science refers to the process by which species evolve and become increasingly distinct from one another over time. It is a result of natural selection and genetic variation within a population.

2. How is divergence different from convergence?

Divergence and convergence are two opposing forces in evolution. While divergence causes species to become increasingly distinct from one another, convergence refers to the process by which unrelated species evolve similar traits in response to similar environmental pressures.

3. What is the role of geographic isolation in divergence?

Geographic isolation, also known as allopatric speciation, is a key factor in divergence. When populations of a species become separated by geographic barriers such as mountains or bodies of water, they are unable to interbreed and may develop distinct traits and characteristics over time.

4. How does divergence contribute to biodiversity?

Divergence is essential for the development of new species and ultimately contributes to biodiversity. As species continue to evolve and diverge, they create new niches and adapt to different environments, leading to a greater variety of life forms on Earth.

5. Can divergence occur within a single species?

Yes, divergence can occur within a single species. This is known as sympatric speciation and is a result of genetic variations and natural selection within a population. Over time, these variations can lead to the formation of distinct sub-species within a single species.

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