# For which values of p and q does an improper integral converge

I am to find for which values of p and q the following integral converges: $$\int_0^\infty \frac{x^p}{1+x^q}\,dx\quad (q>0)$$

As I tested the limit of the above function with $\frac{x^p}{x^q}$, and found it was $1$, I let myself separate the boundaries from $0$ to $1$, and from $1$ to infinity of the later function.

From $0$ to $1$ it's a normal integral, that leaves us to check what the divergence of the integral from $1$ to infinity, which by the comparison test happens as $q>$$1$+$p$

Thanks for the quick replay.

• The integral $\int_0^1\frac{x^p}{1+x^q}\,dx$ is proper, and thus converges for all values of $p,q$. – apnorton Nov 3 '13 at 13:09
• I failed to notice that. That leaves us with q>p+1? – Danny Nov 3 '13 at 13:12
• That should be correct. – apnorton Nov 3 '13 at 13:13
• Thanks. I should edit it. – Danny Nov 3 '13 at 13:14
• I believe that if $p<0$ than that is not true. – Danny Nov 5 '13 at 11:16

Hint : Use the fact that $\zeta(k)=\sum_{n=1}^\infty\frac1{n^k}$ diverges for $k\leqslant1\iff k=q-p>1$ .