$\underset{n\to \infty }{\lim }\sum _{k=1}^n\frac{ k^{{\displaystyle \frac{1}{a}}}\left(n^{a-\frac{1}{a}}+k^{a-\frac{1}{a }}\right)}{n^{a+1}}$ is equal to

The value of $\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{r=0}^{2n-1}\frac{n^2}{n^2+4r^2}$ is:

Let $f:\left(0,2\right)\to R$ be defined as $f\left(x\right)={\log }_2\left(1+\tan \left(\frac{\pi \mathrm{x}}{4}\right)\right)$.

Then, $\underset{n\to \infty }{\lim }\frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\dots .+f\left(1\right)\right)$ is equal to ________.