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

In summary, the conversation discusses strategies for proving convergence or divergence of a function g(x) based on its relationship with another function f(x). The strategies mentioned include using the behavior of rational functions, logarithmic functions, and trigonometric functions to find a comparison. The speaker also mentions struggling with finding the right function g(x) and asks for specific examples to work through.
  • #1
Square1
143
1
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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  • #2
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
 
  • #3
I think that will help well enough. Thank you.
 

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

1. What is Stewart's Integration Strategy?

Stewart's Integration Strategy is a method used to prove the convergence or divergence of an infinite series. It involves evaluating the limit of the sequence of partial sums and comparing it to known convergent or divergent series.

2. How does Stewart's Integration Strategy work?

Stewart's Integration Strategy involves breaking down the given series into smaller, more manageable parts and then using known integration techniques to evaluate each part. The resulting integrals are then compared to known convergent or divergent series to determine the convergence or divergence of the original series.

3. What is the significance of proving g convergence/divergence?

Proving g convergence/divergence is important for determining whether an infinite series is convergent or divergent. This information is crucial in many areas of mathematics and science, including calculus, physics, and engineering.

4. What are some common techniques used in Stewart's Integration Strategy?

Some common techniques used in Stewart's Integration Strategy include partial fraction decomposition, u-substitution, and integration by parts. These techniques help to simplify the integrals and make them easier to evaluate.

5. Are there any limitations to using Stewart's Integration Strategy?

Yes, there are limitations to using Stewart's Integration Strategy. It may not work for all types of series, and the evaluation of the integrals can become increasingly complex for more complicated series. Additionally, it may be difficult to find known convergent or divergent series to compare to in some cases.

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