Let
$\begin{gathered}\lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right. \\ \left.+\cdots \cdots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right) \text { be } \frac{\pi}{k}\end{gathered}$
using only...

Let $\overrightarrow{\mathrm{a}}=2 \hat{\imath}-3 \hat{\jmath}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\imath}+4 \hat{\jmath}-5 \hat{\mathrm{k}}$, and a vector $\vec{c}$ be such that $\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{\imath}+8 \hat{\jmath}+13 \hat{k}$ If $...