Which Method to Use for Testing Convergence in Integrals with Substitution?

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Homework Help Overview

The discussion revolves around testing the convergence of an integral involving the sine function and a square root in the denominator. The integral is presented with limits from 0 to π, and participants are exploring various methods for convergence testing, including integration, direct comparison, and limit comparison tests.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants express uncertainty about which method to apply for testing convergence and question the appropriateness of the Direct Comparison Test. There is also a focus on understanding the limits of the integral and the implications of the denominator approaching zero at the upper limit.

Discussion Status

The discussion is ongoing, with participants clarifying the integral's limits and the form of the integrand. Suggestions have been made regarding the use of substitution to simplify the problem, and there is an exploration of potential inequalities for comparison.

Contextual Notes

There is a noted concern about the denominator becoming zero at the upper limit of the integral, which raises questions about the convergence of the integral. Participants are also navigating the lack of detailed instruction from the instructor regarding method selection.

heal
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Homework Statement


Use integration, the direct comparison test, or the limit comparison test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.


Homework Equations


∫sinθdθ/√π-)


The Attempt at a Solution


I don't know which method to use and why in this case why I would apply one of them. Our instructor has not gone through in detail which methods for which cases. Thank you.
I'm guessing that the Direct Comparison Test would be useful here. Though, I am not sure what equation to use as a comparison.
 
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heal said:

Homework Statement


Use integration, the direct comparison test, or the limit comparison test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.


Homework Equations


∫sinθdθ/√π-)

What are the limits on the integral? What is that supposed to be after the / ?
 
LCKurtz said:
What are the limits on the integral? What is that supposed to be after the / ?
The limits are from 0 to π.

After the / it's "rad(π - θ)


Sorry.
 
So it is$$
\int_0^\pi \frac {\sin \theta}{\sqrt{(\pi -\theta)}}\, d\theta$$

The problem is the denominator is 0 when ##\theta = \pi##. Your problem is to figure out whether that makes the integral diverge or not. Of course, the numerator is 0 there too, so it could be either way. While it isn't absolutely necessary, still I would suggest the substitution ##u = \pi - \theta## to simplify it and move the difficulty to ##u=0##. Then see what you think. You might find some inequality to try for comparison.
 

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