Magical test for convergence of improper integrals?

Question

I found this article while surfing the web. I hope it's not some kind of joke, because if it is it really fooled me. I'm trying to figure out the proof of theorem 2.3

I don't understand how the author went from $\lim_{x \to \infty}xR(x) > 1$ to:

$$\forall k>0, \space xR(x) \ge k+1$$

Thanks for the help.

Answer 1

Is false: $R(x)=2/x$ verifies $$\lim_{x\to\infty}xR(x)=2>1$$ but $$\forall k: 2=xR(x)> k+1$$ is false.

EDIT:

Barely understandable phrases: "But it is difficult or impossible to find the comparison infinite." "It is well-known that the convergence and divergence of infinite integral for the different integrand which its limit is zero is different."

Weird expressions: "incredibly similar".

Wrong application of L'Hôpital in the proof of Theorem 2.1.

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