Let lim_n → ∞(n/(√(n^4+1))-(2 n)/((n^2+1) √(n^4+1))+n/(√(n^4+16))-(8 n)/((n^2+4) √(n^4+16))..+…+n/(√(n^4+n^4))-(2 n · n^2)/((n^2+n^2) √(n^4+n^4))) be…

Let $\displaystyle \lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$ $\displaystyle \left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$ be $\displaystyle \frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $\displaystyle \mathrm{k}^2$ is equal to $\displaystyle \_\_\_\_$.