The value of $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^{\frac{1}{6}}}{\sqrt[6]{k^4 + n^4}}$ is equal to
Step-by-Step Solution
Key Concept: Recognizing derivative forms and applying limit theorems to evaluate limits of complex expressions involving inverse trigonometric functions.
$\lim_{x \to \infty} \frac{\pi}{2x} \left(\frac{1}{\sqrt{1-\left(1-\frac{1}{x}\right)^2}}\right) = \frac{1}{2}$. This limit is evaluated using L'Hôpital's rule or by recognizing the form of the arcsine derivative.
Correct Answer: 1