I'm a bit confused about how to use this test. So the theorem for it basically says, if the limit as x -> inf. f(x) / g(x) =L and 0<L<inf.
then the integral of f(x) and the integral of g(x) both converge or diverge.
so if i have a particular function that i need to find whether it converges or diverges, and i want to use this test, when i pick another function to compare it to, do i have to pick one that i already know whether or not it converges?
an example that is shown in my book is:
integral dx/(1+x^2) (and it's from 1 to inf.) so the function is just 1/(1+x^2).
they choose to compare it to 1/x^2 because the two functions are essentially the same at infinity. but that's what is confusing me. so should i choose a function that is similar to the given one at infinity, or just pick a function that i already know if it converges or diverges, or both? i think it's a crappy example.