Let ` f_(p) ( beta) = ( cos "" (beta)/( p^(2)) + i sin "" (beta)/( p^(2))) ( cos ""( 2 beta)/( p^(2)) + i sin "" ( 2 beta)/( p^(2))) . . . (cos "" (beta)/( p) + i sin "" (beta)/(p))` then ` lim_(n to infty) f_(n) (pi)` =

Options

A. i
B. `-i`
C. 2i
D. `-2i`

Answer

To solve the problem, we need to evaluate the limit of the function \( f_n(\beta) \) as \( n \) approaches infinity, specifically for \( \beta = \pi \). Let's break it down step by step.

### Step 1: Define the function
The function is given as:
\[ f_p(\beta) = \left( \cos\left(\frac{\beta}{p^2}\right) + i \sin\left(\frac{\beta}{p^2}\right) \right) \left( \cos\left(\frac{2\beta}{p^2}\right) + i \sin\left(\frac{2\beta}{p^2}\right) \right) \cdots \left( \cos\left(\frac{\beta}{p}\right) + i \sin\left(\frac{\beta}{p}\right) \right) \]
This can be rewritten using Euler's formula:
\[ f_p(\beta) = e^{i \frac{\beta}{p^2}} e^{i \frac{2\beta}{p^2}} \cdots e^{i \frac{\beta}{p}} \]

### Step 2: Combine the exponentials
We can combine the exponentials:
\[ f_p(\beta) = e^{i \left( \frac{\beta}{p^2} + \frac{2\beta}{p^2} + \cdots + \frac{\beta}{p} \right)} \]
The sum in the exponent can be simplified. The series is:
\[ \frac{\beta}{p^2} + \frac{2\beta}{p^2} + \cdots + \frac{p\beta}{p^2} \]
This can be factored as:
\[ \frac{\beta}{p^2} (1 + 2 + \cdots + p) \]

### Step 3: Calculate the sum of the first \( p \) natural numbers
The sum of the first \( p \) natural numbers is given by:
\[ 1 + 2 + \cdots + p = \frac{p(p + 1)}{2} \]
Thus, we have:
\[ f_p(\beta) = e^{i \frac{\beta}{p^2} \cdot \frac{p(p + 1)}{2}} = e^{i \frac{\beta (p + 1)}{2p}} \]

### Step 4: Substitute \( \beta = \pi \) and evaluate the limit
Now, we need to find:
\[ \lim_{n \to \infty} f_n(\pi) = \lim_{n \to \infty} e^{i \frac{\pi (n + 1)}{2n}} \]
As \( n \to \infty \), \( \frac{n + 1}{2n} \) approaches \( \frac{1}{2} \). Therefore:
\[ \lim_{n \to \infty} f_n(\pi) = e^{i \frac{\pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) = 0 + i \cdot 1 = i \]

### Final Answer
Thus, the limit is:
\[ \lim_{n \to \infty} f_n(\pi) = i \]