Question

$\lim_{n \to \infty} \left(\frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \dots + \frac{n}{n^2 + 3^2 + \dots + \frac{1}{5n}} \right) =$

• A. $\frac{\pi}{4}$
• B. $\tan^{-1} 3$
• C. $\tan^{-1} 2$
• D. $\frac{\pi}{2}$

Answer 1

The Correct Option is C

Rewrite the given sum as: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n^2 + k^2}. \] This simplifies to a Riemann sum. By integrating and using standard trigonometric limits, we get: \[ \lim_{n \to \infty} = \tan^{-1} 2. \]