The value of `lim_(nrarroo)(1)/(n)[sec^(2)""(pi)/(4n)+sec^(2)""(2pi)/(4n)+….+sec^(2)""(npi)/(4n)]` is

Options

A. `log_(e)2`
B. `(pi)/(2)`
C. `(4)/(pi)`
D. e

Answer

To solve the limit

\[ \lim_{n \to \infty} \frac{1}{n} \left[ \sec^2 \left( \frac{\pi}{4n} \right) + \sec^2 \left( \frac{2\pi}{4n} \right) + \ldots + \sec^2 \left( \frac{n\pi}{4n} \right) \right], \]

we can express the sum as a Riemann sum, which will help us convert it into a definite integral.

### Step-by-Step Solution

1. **Rewrite the limit as a summation**:
We can express the limit as:
\[ \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \sec^2 \left( \frac{r\pi}{4n} \right). \]

2. **Identify the Riemann sum**:
As \( n \to \infty \), the term \( \frac{r}{n} \) can be thought of as a variable \( x \) ranging from 0 to 1. We can rewrite \( \frac{r\pi}{4n} \) as \( \frac{\pi}{4} \cdot \frac{r}{n} = \frac{\pi}{4} x \). Thus, we can express the summation as:
\[ \lim_{n \to \infty} \sum_{r=1}^{n} \sec^2 \left( \frac{\pi}{4} \cdot \frac{r}{n} \right) \cdot \frac{1}{n}. \]
This is a Riemann sum for the function \( \sec^2 \left( \frac{\pi}{4} x \right) \) over the interval [0, 1].

3. **Convert to a definite integral**:
Therefore, we can write:
\[ \lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \sec^2 \left( \frac{r\pi}{4n} \right) = \int_{0}^{1} \sec^2 \left( \frac{\pi}{4} x \right) dx. \]

4. **Evaluate the integral**:
The integral can be evaluated using the known integral of \( \sec^2 x \):
\[ \int \sec^2 x \, dx = \tan x + C. \]
Thus,
\[ \int_{0}^{1} \sec^2 \left( \frac{\pi}{4} x \right) dx = \left[ \tan \left( \frac{\pi}{4} x \right) \cdot \frac{4}{\pi} \right]_{0}^{1}. \]

5. **Calculate the limits**:
- At \( x = 1 \):
\[ \tan \left( \frac{\pi}{4} \cdot 1 \right) = \tan \left( \frac{\pi}{4} \right) = 1. \]
- At \( x = 0 \):
\[ \tan \left( \frac{\pi}{4} \cdot 0 \right) = \tan(0) = 0. \]
Therefore,
\[ \int_{0}^{1} \sec^2 \left( \frac{\pi}{4} x \right) dx = \left( 1 \cdot \frac{4}{\pi} \right) - \left( 0 \cdot \frac{4}{\pi} \right) = \frac{4}{\pi}. \]

6. **Final result**:
Thus, the value of the limit is:
\[ \frac{4}{\pi}. \]