I am working with some comparison testing for the first time, and am unsure if the method I have adopted is legit.
For example, imagine I want to compare some improper integrals (or series) with the integral (or series) of $\frac{1}{x^P}$. We know this has some nice properties for $P > 1$ and $P < 1$. However, when comparing with some function, I also need that limit to behave accordingly, going to a finite value when I need it to, and going to something bigger than zero when I need it to.
So what I do is that I just take, say, some value $b \in (1,P)$ (in the case where $P > 1$), and then compare with $\frac{1}{x^b}$ instead. Then, I get $$\frac{f(x)}{g(x)} = \text{'stuff'} x^{P - b} \text{'stuff'}$$ which blasts of towards infinity as long as "stuff" is under control. Similarly, when $P < 1$, I'd take $b \in (P,1)$, and get something that goes towards zero. Now of course as $b < 1$ and $b > 1$ in both these cases, the series or integral related to $\frac{1}{x^b}$ converges and diverges in these cases... Is this method okay, or is there some flaw in it?
Please tell me if it was not clear enough, then I'll dig up an example where I used it and post it. I just don't have that at hand right now.
