If $x_{n}=cos\frac{\pi}{2^{n}}+i~sin\frac{\pi}{2^{n}}$; $n\in N,$ then $x_{1}x_{2}x_{3}$........... upto $\infty$ is equal to:

Question

Accepted Answer

To solve this problem, we need to evaluate the infinite product of complex numbers given by $ x_n = \cos\left(\frac{\pi}{2^n}\right) + i \sin\left(\frac{\pi}{2^n}\right) $. This expression can be rewritten using Euler's formula as:

$$ x_n = e^{i \frac{\pi}{2^n}} $$

We are tasked with finding the value of the infinite product:

$$ \prod_{n=1}^{\infty} x_n = \prod_{n=1}^{\infty} e^{i \frac{\pi}{2^n}} $$

Using the property of exponents, the product of exponentials can be expressed as the exponential of the sum of the exponents:

$$ \prod_{n=1}^{\infty} e^{i \frac{\pi}{2^n}} = e^{i \sum_{n=1}^{\infty} \frac{\pi}{2^n}} $$

The series $\sum_{n=1}^{\infty} \frac{\pi}{2^n}$ is a geometric series with the first term $a = \frac{\pi}{2}$ and common ratio $r = \frac{1}{2}$. The sum of an infinite geometric series is given by:

$$ S = \frac{a}{1 - r} = \frac{\frac{\pi}{2}}{1 - \frac{1}{2}} = \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi $$

Thus, the sum of the series is $\pi$, and the product becomes:

$$ e^{i \pi} $$

According to Euler's formula, $ e^{i \pi} = \cos(\pi) + i \sin(\pi) = -1 + 0i = -1 $.

Therefore, the infinite product evaluates to $-1$, which corresponds to Option 3.

### Explanation of Incorrect Options:

1. **Option 1 (0):** The product of complex numbers in this form cannot be zero unless one of the terms is zero, which is not the case here. Each $ x_n $ is a unit complex number on the unit circle in the complex plane, so their product cannot be zero.

2. **Option 2 (1):** The product evaluates to $ e^{i \pi} = -1 $, not 1. The value 1 would be the result if the sum of the exponents were a multiple of $2\pi$, which is not the case here.

3. **Option 4 ($\infty$):** The product of unit complex numbers cannot diverge to infinity. Each $ x_n $ has a magnitude of 1, so the product remains bounded.

In conclusion, the correct answer is Option 3, $-1$, as derived from the evaluation of the infinite product using Euler's formula and the properties of geometric series.