1)Show that for all $n\in\mathbb{N}$ the following is true:
$$\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$$
for a constant $C>0$ and conclude that the improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ isn't absolutely convergent.
2)Show that the improper integral $\int_0^\infty \frac{1-\cos(x)}{x^2}dx$ is absolutely convergent. (The integrand is to be expanded continuous at $x=0$.).
3)Using 2), show that the improper integral $\int_0^\infty \frac{\sin(x)}{x}dx$ is convergent.
We started discussing improper integrals in class and our prof showed us how some can be solved and some can't.
Anyways,
Here were my ideas so far:
1) I thought about to do the integral and seeing if what I get out of it gives me any idea to show the inequality. But I couldn't even solve the integral (not by hand nor with the help of an integral calculator). So I don't know what to do next.
2)To be hoenst I'm totally lost here. No idea how to approach it.
3)Well, since I didn't solve 2).
Sorry for my lack of work here, but this topic just doesn't want to stick with me.