Question

Find the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) = \]

• A. \( \frac{2\pi}{5} \)
• B. \( \frac{5}{2} \)
• C. \( \frac{2}{5} \)
• D. \( \frac{5\pi}{2} \)

Hint:

To solve a Riemann sum, convert it to the corresponding integral and compute the integral.

Answer 1

The Correct Option is C

Step 1: Recognizing the integral form.
The given sum is a Riemann sum for the integral of the function \( \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot x \right) \) over the interval [0, 1]. The Riemann sum approximation for large \( n \) is given by: \[ \sum_{k=1}^{n} \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot \frac{k}{n} \right) \approx n \int_0^1 \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot x \right) dx. \]
Step 2: Solving the integral.
The integral is straightforward: \[ \int_0^1 \sin \left( \frac{\pi}{2} + \frac{5\pi}{2} \cdot x \right) dx. \] Using the substitution \( u = \frac{5\pi}{2} \cdot x + \frac{\pi}{2} \), the integral evaluates to \( \frac{2}{5} \).

Step 3: Conclusion.
The limit of the sum is \( \frac{2}{5} \). Thus, the correct answer is (C) \( \frac{2}{5} \).