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tnutty
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Homework Statement
Determine whether the series is convergent or divergent.
[tex]\sum[/tex] n5 / (n6 + 1)
tnutty said:Yes I was thinking of the comparison test, but that's next chapter. in this chpt, its all about integral test. i am not sure how to solve this with integral test, but can you check out the comparison test that follows ?
Comparison test ;
n^5 / (n^6+1) <= n^5 / n^6 = 1/n and from definition we know that 1/(n^p)
converges if n > 1 and diverges if n< 1. So in this case it diverges since n = 1.
?
Sorry to burst your bubble, but no it does not. Take a look at the comparison test and what it says about divergent series and what it says about convergent series. They are different.rwisz said:The comparison test does show divergence that's right.
rwisz said:For the integral test however, since the numerator contains [tex]n^5[/tex] and the derivative of the denominator is [tex]6n^5[/tex] then you should be able to tell that u-substitution will work like a charm here...
Hint: du/u = ln u.
And for the setup of the improper integral, try looking at the previous thread where I helped you, at the bottom of my last post.
Mark44 said:Sorry to burst your bubble, but no it does not. Take a look at the comparison test and what it says about divergent series and what it says about convergent series. They are different.
"Convergent and divergent" refers to two types of evolution or adaptation in species. Convergent evolution is when two unrelated species develop similar traits in response to similar environmental pressures. Divergent evolution is when a single species evolves into two or more distinct species due to different environmental pressures.
Scientists study convergent and divergent evolution through a variety of methods, including comparing DNA sequences, examining fossil records, and conducting experiments in controlled environments. They also use mathematical models and computer simulations to understand the processes and patterns of evolution.
An example of convergent evolution is the similarities between the wings of birds, bats, and insects, which all evolved independently to allow for flight. An example of divergent evolution is the different beak shapes in finches on the Galapagos Islands, which evolved to suit different food sources and environments.
Convergent and divergent evolution play important roles in creating and maintaining biodiversity. Convergent evolution leads to the development of new species with similar traits, while divergent evolution leads to the development of distinct species with unique traits. This diversity allows for species to adapt to changing environments and increases the overall resilience of ecosystems.
Understanding convergent and divergent evolution can help scientists and medical professionals better understand how diseases evolve and spread, and how to develop more effective treatments. It can also provide insights into the development of new drugs and technologies through the study of convergent evolution in plants and animals.