$$ \int_1^{\infty} \left\langle t\right\rangle\dfrac{\cos\left(t\right) - \sin\left(t\right)}{t^2}\,dt $$
where $\left\langle t\right\rangle$ is the rationale part of $t$.
I would like to use the improper integral comparison test with $ \dfrac 2{t^2} $, but I have no reason to think the integral is positive. How can I evaluate if this function is integrable in the improper sense? Thank you.
Don't worry about $\cos t - \sin t$, because $$ \left|\langle t\rangle \frac{\cos t - \sin t}{t^2}\right| \le \frac{2}{t^2} $$ and so we can apply absolute convergence of improper integral.