JEE Previous Year Question

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

Topic: JEE PYQ

Answer: 32. Full step-by-step solution below.