Find the convergence region of the improper integral $\int_1^\infty \frac{x^a}{1+x^a}dx$

Question

Find the convergence region of the improper integral $$\int_1^\infty \frac{x^a}{1+x^a}dx\,.$$

To find the convergence region of this integral, it is equivalent to find the convergence region of the series $$\sum_{n=1}^\infty \frac{n^a}{1+n^a}\,.$$

I use the ratio test and root test but could find the convergence area for $a$.

Answer 1

The integral is convergent for $-a \gt 1$ or $a\lt-1$ and otherwise it is divergent.