By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent.
- Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{n^\alpha}\mbox{ is convergent}\right\}?$$
- In the case of a positive answer to the previous question, what is $$\inf\left\{\beta\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{\sqrt{n}(\log n)^\beta}\mbox{ is convergent}\right\}?$$
This is a generalized problem, so I don't think "try a lot of numbers" is a correct way.
Area of the circle: $\pi r^2$
Area of the square: $a^2$
So we need $\pi r^2=a^2$ $\Leftrightarrow$ $\dfrac{a^2}{r^2}=\pi$ $\Leftrightarrow a=r\sqrt{\pi }$