# Improper convergence of this integral?

$$\int_1^{\infty} \left\langle t\right\rangle\dfrac{\cos\left(t\right) - \sin\left(t\right)}{t^2}\,dt$$

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.

• Two hints: $\langle t \rangle < 1$ and absolute convergence implies conditional convergence! – Nigel Overmars Mar 10 '16 at 11:20

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.