Is $\int_0^{\infty} \frac{t^4}{1+t^6} dt$ convergent?

Question

Question: Is $\int_0^{\infty} \frac{t^4}{1+t^6} dt$ convergent?

Try: I am trying to apply limit compression test but I am unable to decide ..

Answer 1

Hint. A potential problem is as $ t \to \infty$, but one may observe that $$ 0\le\frac{t^4}{1+t^6}\le \frac{1}{t^2}, \qquad t\ge1, $$ giving the convergence of the given integral by comparison to a Riemann integral $$ \int_1^\infty \frac{dt}{t^\alpha}<\infty,\qquad \alpha>1. $$