Question

If \(x_{n}=\cos \left(\pi / 2^{n}\right)+i \sin \left(\pi / 2^{n}\right)\) then \(x_{1} x_{2} x_{3} \ldots \ldots \infty=\) (a) - i (b) \(-1\) (c) (d) 1

Short answer

The product of the infinite sequence \(x_1x_2x_3\ldots\) is equal to \(-1\). The correct answer is (b) \(-1\).

Step-by-step solution

Identify Euler's formula

Euler's formula is given as: \[e^{ix} = \cos x + i\sin x\]

Rewrite the expression using Euler's formula

Using Euler's formula, we can rewrite the expression for \(x_n\) as: \[x_n = e^{i\left(\pi / 2^{n}\right)}\]

Find the product of the sequence

We are looking for the product of the infinite sequence \(x_1x_2x_3\ldots\), which can be represented as the following: \[x_1x_2x_3\ldots = e^{i\left(\frac{\pi}{2^1}\right)} \cdot e^{i\left(\frac{\pi}{2^2}\right)} \cdot e^{i\left(\frac{\pi}{2^3}\right)}\ldots\]

Combine exponents with the same base

As the base of the exponential function is consistent, we can combine the exponents: \[x_1x_2x_3\ldots = e^{i\left(\frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + \ldots\right)}\]

Find the sum of the geometric series

The exponent is a geometric series with a first term of \(\frac{\pi}{2}\) and a common ratio of \(\frac{1}{2}\). The sum of this series can be calculated using the formula: \[\frac{\text{first term}}{1 - \text{common ratio}}\] So, the sum of the series is given by: \[\frac{\frac{\pi}{2}}{1 - \frac{1}{2}} = \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi\]

Substitute the sum back into the expression and simplify

Now, substitute the sum of the geometric series back into the expression for the product of the infinite sequence: \[x_1x_2x_3\ldots = e^{i\pi}\] Using Euler's formula, we know that \(e^{i\pi} = \cos\pi + i\sin\pi\), and as \(\cos\pi = -1\) and \(\sin\pi = 0\), we have: \[x_1x_2x_3\ldots = -1\] Therefore, the product of the infinite sequence is equal to: \[\boxed{-1}\] The correct answer is (b) \(-1\).

Key concepts

Euler's Formula

Euler's formula provides a fascinating connection between complex numbers and exponential functions. It is generally expressed as: \[ e^{ix} = \cos x + i\sin x \] This relationship is incredibly useful in mathematics, especially when dealing with complex numbers in polar form. Rather than expressing a complex number using standard Cartesian coordinates (real and imaginary parts), Euler's formula allows us to use an exponential form. This simplifies many operations, especially multiplication and powers of complex numbers. By using Euler's formula, many problems that involve trigonometric functions and complex numbers become easier to solve. In the context of the original exercise, we utilized Euler's formula to transform the expression for \(x_n\) into an exponential form, making it more amenable to evaluation.

Geometric Series

A geometric series is a sequence of numbers where each term is fixed, repeated multiplication, or division of the previous term by a constant, known as the common ratio. The general form of a geometric series can be written as:

• First term: \(a\)
• Common ratio: \(r\)

The sum \(S\) of an infinite geometric series, where \(|r| < 1\), is given by the formula: \[ S = \frac{a}{1 - r} \] In the original exercise, the exponent of the combined exponential product forms a geometric series with a first term \(\frac{\pi}{2}\) and a common ratio of \(\frac{1}{2}\). By calculating the sum of this series, we can move closer to finding the product of the complex terms in the problem, leading us to a neat conclusion. Understanding how to sum a geometric series is crucial in evaluating such expressions efficiently.

Infinite Product

An infinite product involves multiplying a sequence of terms together endlessly. Mathematically, it is expressed as the product of terms in the form \(a_1 \cdot a_2 \cdot a_3 \cdot \ldots\). This concept is particularly significant when addressing the convergence of the product. For a product to be finite and convergent, similar considerations are applied as in infinite series. That is, the terms must approach a significance that allows the series or product to stabilize.In the original problem, the sequence \(x_1x_2x_3\ldots\) represents an infinite product of complex exponential terms. By using properties of exponentials, specifically the ability to sum the exponents when multiplying like bases, we can simplify the infinite product to a manageable form. This leads us to discover the ultimate product of the sequence is \(-1\).

Exponential Function

The exponential function \(e^x\) is a remarkable mathematical construct, defined as: \[ e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n \] This function is crucial for growth processes, complex numbers, and Euler's identity and formula. Its unique property is the rate of growth proportional to its size.When we extend this to the complex plane, things get even more interesting. The expression \(e^{ix}\) incorporates Euler's formula, creating a bridge between exponential calculations and trigonometric functions. In the context of the problem you're solving, converting the original expression into an exponential form with the base \(e\) simplified the infinite product calculation. Thus, grasping the exponential function's role is essential for navigating complex mathematical landscapes.