Comparing Stewart's Integration Strategy: Proving g Convergence/Divergence

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SUMMARY

This discussion focuses on Stewart's integration strategy for proving convergence and divergence of functions. The key takeaway is that when g(x) is greater than or equal to f(x) and both are non-negative, one should prove convergence on g. Conversely, if f(x) is greater than or equal to g(x), divergence on g should be established. The discussion emphasizes the importance of selecting the correct function g to avoid incorrect conclusions about f(x), highlighting that rational functions are dominated by their highest degree terms, while logarithmic functions grow slower than polynomial functions, which in turn grow slower than exponential functions.

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Square1
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Just as talked about in stewart in strategy for integration.

I found notes online that also say:

g(x) >= f(x) >= 0, then you want to prove convergence on g. If f(x) >= g(x) >= 0, then you want to prove divergence on g. I am pretty sure I follow the logic here, but how exactly does one pick g?? I've been working on hours picking g that results in the opposite of required case (ex. divergence for g(x) >= f(x) >= 0) which does not then prove anything about f(x)! :(
 
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The typical strategies for simple problems are to note that rational functions behave like their highest degree terms, log grows slower than power functions which grow slower than exponentials, and sine and cosine are bounded by one. If you have some specific examples in mind we can work through finding the comparison
 
I think that will help well enough. Thank you.
 

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