For which values of $r$ and $s$ does $ \int_{0}^{2} \frac{(1+x)^r-(1+x)^s}{x^2} dx $ converge?

Question

My guess is that r and s need t be large enough to make sure that the numerator is bigger, in a sense, than x, in order to ensure that the denominator has an x with a power that is smaller, in a sense, than 1, in $$ \int_{0}^{2} \frac{(1+x)^r-(1+x)^s}{x^2} dx. $$ I find it very difficult, to make this precise though.

Answer 1

Near $0$, we have

$$(1+x)^p=1+px (1+\epsilon(x)) $$ with $$\lim_{x\to0}\epsilon (x)=0.$$

thus your function is equivalent to

$$\frac {r-s}{x} $$ the integral is divergent. it converges only if $r=s $.