The question asks whether the following converges or diverges.
$$
\int_{0}^{\infty }
{\left\vert\,\sin\left(\,x\,\right)\,\right\vert \over x^2}\,{\rm d}x
$$
Now I think there might be a trick with the domain of sine function but I couldn't make up my mind on this.
I tried to compare it with $1/x^{2}$, $\sin\left(\,x\,\right)/x$, and
$\sin\left(\,x\,\right)$.
I actually expected that something good would come from $1/x^{2}$, but as the lower limit of the integral is zero, it ended up with infinity on $\left(\,0,\infty\,\right)$.
Since $1/x^{2}$ is greater than the given function, and is divergent on the given interval, it doesn't help at all.
So I'm wondering what is the right track on this problem ?.
Best Answer
$\frac{x}{2}<|sin(x)|$ on the interval of $(0,\pi /2)$. Therefore $\frac{|sin(x)|}{x^2} >\frac{\frac{x}{2}}{x^2}=\frac{1}{2x}$.
since $\int_0^{\pi /2} \frac{1}{2x}$ doesn't converge to a positive real number, $\int_0^{\pi /2} \frac{|sin(x)|}{x^2} $ won't converge to a real number.