Can you, please, check if it's right what I did:
Here's the exercise:
Test the convergence of the following improper integral which is defined using parameter $p\in R$: $$\int_{0}^{\infty}\frac{x^p}{1 + x}$$
Here's what I thought of:
I used, a criteria that translated in english would be: Alpha criterium (I could not find the proper english name)
So, I took $\alpha = 1 - p$
Case 1:
If $\alpha \in (1, \infty)$ and $\lim_{x\to \infty}{x^{\alpha}\frac{x^p}{1+x}} = 1$, where $0 \le 1 < \infty$ then the integral is convergent.
Case 2:
If $\alpha \in (\infty, 1]$ and $\lim_{x\to \infty}{x^{\alpha}\frac{x^p}{1+x}} = 1$, where $0 < 1 \le \infty$ then the integral is divergent.
So:
For $p < 0$ the integral is convergent, and for $p \ge 0$ the integral is divergent.