1. ## Improper Integral

The initial problem is to find whether

from 0 to $\displaystyle \pi$ $\displaystyle \int\frac{7 - sin(\alpha)}{\alpha^2}$ converges or diverges. I found that it diverges.

Then the second part of the problem asks:

Which of the following inequalities can be used to help prove your conclusion?

I think it's the first option because the integral I'm comparing to has to be bigger than the original one for it to have any meaning, but should it be

$\displaystyle \frac{6}{\alpha^2}$?

Or is that entirely wrong? I have no idea how to determined which inequality it should be.

Thank you for any help, and if you could explain a little of how you determined which one it was, I'd really appreciate it.

2. just a limit comparison test with $\displaystyle \int_0^\pi\frac{d\alpha}\alpha.$

3. I don't understand what that means, sorry. Could you explain?

4. have you ever taught the limit comparison test? it's pretty useful when you do not find the appropriate inequality for your integral.

5. Is that where you replace infinity with "b" and take the limit as b goes to infinity of the integral? If so, yes I know that. That's partially how I determined that the integral diverges.

I just don't understand the second part...