if x r=cos(pi/2^r)+i sin(pi/2^r),then x1*x2*x3*x4....infinity=?

To tackle the problem of finding the product of the series $x_1,x_2,x_3,\dots$ where $x_r=\cos \left(\frac{\pi }{2^r}\right)+i\sin \left(\frac{\pi }{2^r}\right)$, we can start by recognizing that this is a complex exponential form. According to Euler's formula, we know that $e^{i\theta }=\cos (\theta )+i\sin (\theta )$. Therefore, we can express each term $x_r$ as:

Transforming the Terms

Using Euler's formula, we write:

x_r = e^{i \frac{\pi}{2^r}}

Understanding the Infinite Product

We want to find the infinite product:

x_1 \cdot x_2 \cdot x_3 \cdots

Substituting our expression for $x_r$ into the product gives:

x_1 \cdot x_2 \cdot x_3 \cdots = e^{i \frac{\pi}{2^1}} \cdot e^{i \frac{\pi}{2^2}} \cdot e^{i \frac{\pi}{2^3}} \cdots

Combining Exponents

When multiplying exponentials, we can add the exponents:

e^{i \left( \frac{\pi}{2^1} + \frac{\pi}{2^2} + \frac{\pi}{2^3} + \cdots \right)}

Calculating the Series

Next, we need to evaluate the series:

$\frac{\pi }{2^1}+\frac{\pi }{2^2}+\frac{\pi }{2^3}+\cdots$

This is a geometric series where the first term $a=\frac{\pi }{2}$ and the common ratio $r=\frac{1}{2}$. The sum $S$ of an infinite geometric series can be calculated using the formula:

S = \frac{a}{1 - r}

Plugging in our values:

S = \frac{\frac{\pi}{2}}{1 - \frac{1}{2}} = \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi

Final Result

Thus, we find that:

x_1 \cdot x_2 \cdot x_3 \cdots = e^{i \pi}

According to Euler's identity, $e^{i\pi }=-1$. Therefore, the product of the series is:

-1

Summary

In conclusion, the infinite product $x_1\cdot x_2\cdot x_3\cdots$ evaluates to $-1$. This demonstrates not only the power of series and exponential functions but also the beauty of how complex numbers can simplify certain mathematical expressions.