Find the parameter values for which the integral $\int_1^\infty \frac{dx}{x^p - x}$ converges.

Question

I was trying to solve this question but I was really stuck on trying to find a function to compare the integrand of this integral to. Can anyone help me solve this using the Convergence Comparison Test?

$$\int_1^\infty \frac1{x^p - x}dx$$

For which p values does this converge, and how do you prove it? The correct answer should be for no p.

Answer 1

Hint:$$\lim_{x\to1^+}\frac{\frac1{x^p-x}}{\frac1{x-1}}=\lim_{x\to1^+}\frac {x-1}{x^p-x}=\frac1{p-1}.$$