Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that $$ \lim _{n \rightarrow \infty} \cos \frac{x}{2} \cos \frac{x}{4} \cos \frac{x}{8} \cdots \cos \frac{x}{2^{n}}=\frac{\sin x}{x} $$ by completing the following steps. $$ \begin{array}{l} \text { (a) } \cos \frac{x}{2} \cos \frac{x}{4} \cdots \cos \frac{x}{2^{n}}= \\\ {\left[\cos \frac{1}{2^{n}} x+\cos \frac{3}{2^{n}} x+\cdots+\cos \frac{2^{n}-1}{2^{n}} x\right] \frac{1}{2^{n-1}}} \end{array} $$ (See Problem 46 of Section 1.8.) (b) Recognize a Riemann sum leading to a definite integral. (c) Evaluate the definite integral.

Step-by-Step Solution

Answer

The limit equals \(\frac{\sin x}{x}\).

Step 1: Expression Setup

Start with the given infinite product of cosine terms: \[ \prod_{k=1}^{n} \cos \frac{x}{2^k}. \] Recognize that we need to manipulate and simplify this expression based on part (a) of the problem.

Step 2: Simplifying Using Sum of Cosines

Recall that the product formula given is: \[ \cos \frac{x}{2} \cos \frac{x}{4} \cdots \cos \frac{x}{2^n} = \frac{1}{2^{n-1}} \left( \cos \frac{1}{2^n} x + \cos \frac{3}{2^n} x + \cdots + \cos \frac{2^n-1}{2^n} x \right). \] This formula expresses the product of cosines as a sum of cosines.

Step 3: Recognizing a Riemann Sum

Notice that the expression \[ \frac{1}{2^{n-1}} \left( \cos \frac{1}{2^n} x + \cos \frac{3}{2^n} x + \cdots + \cos \frac{2^n-1}{2^n} x \right) \] can be rewritten to recognize a Riemann sum, where \(\frac{x}{2^n}\) plays the role of the width of subintervals in the context of integration from \(0\) to \(x\).

Step 4: Evaluate the Riemann Sum

As \(n\) approaches infinity, the sum becomes a Riemann sum for the integral: \[ \int_0^x \cos t \, dt. \] The limit of the sum as \(n\) goes to infinity approximates this definite integral.

Step 5: Compute the Definite Integral

The indefinite integral of \( \cos t \) is \( \sin t \). Thus, \[ \int_0^x \cos t \, dt = \sin x - \sin 0 = \sin x. \] So the limit of the original product equals \(\frac{\sin x}{x}\).

Key Concepts

Riemann Sum

The idea of a Riemann sum is central to understanding the relation between a sum of areas and the integral of a function. In a Riemann sum, the integral of a function is approximated by dividing the area under a curve into a finite number of rectangles. Each rectangle's height comes from the function's value at a specific point within that rectangle, and its width corresponds to a certain subinterval of the entire interval.

• The interval \( [a, b] \) is divided into n subintervals of equal width.
• Each width is given by \( \Delta x = \frac{b-a}{n} \).
• The sum is computed as \( \sum_{i=1}^{n} f(x_i^*) \Delta x \), where \( x_i^* \) is a sample point in each subinterval.

The connection to calculus becomes apparent as the number of rectangles increases, leading to more accurate approximations.
The limit of a Riemann sum as the number of subintervals approaches infinity is called the definite integral of the function, providing us with the exact area under the curve.

Definite Integral

The definite integral, denoted as \( \int_a^b f(x) \, dx \), represents the net area between the function \( f(x) \) and the x-axis from \( a \) to \( b \). It provides a precise mathematical tool to find such areas and helps in various applications in physics, economics, and beyond.

• The Fundamental Theorem of Calculus links the indefinite integral and the definite integral, showing how antiderivatives can be used to evaluate definite integrals.
• The definite integral is calculated using limits of sums, specifically through the technique of Riemann sums.
• In practice, we use antiderivatives to find \( F(b) - F(a) \), where \( F \) is an antiderivative of \( f \).

For the function \( \cos t \), the integral over \( [0, x] \) results in \( \sin x - \sin 0 = \sin x \), illustrating how the areas under trigonometric functions are determined.

Cosine Function

The cosine function, often written as \( \cos(x) \), is a key trigonometric function in mathematics, describing the x-coordinate of the point on the unit circle.
It is periodic with a period of \( 2\pi \), and its values oscillate between -1 and 1. The function has numerous properties:

• It is even, meaning \( \cos(-x) = \cos(x) \).
• The cosine of 0 is 1, which is its maximum value.
• The values at \( \pi/2 \) and \( 3\pi/2 \) are 0, reaching its minimum value at \( \pi \).

These properties make cosine a fundamental tool in modeling waves, oscillations, and circles. Additionally, due to its role in triangle measurements, \( \cos(x) \) links geometry with algebraic calculation, enriching its application scope.

Infinite Product

An infinite product refers to the multiplication of an infinite number of terms, often represented as \( \prod_{k=1}^{\infty} a_k \), where each term in the sequence is multiplied together.
In calculus and analysis, understanding the behavior of infinite products helps in exploring convergence and evaluating limits in special series or products.

• An infinite product converges to a limit if the partial products \( P_n = \prod_{k=1}^{n} a_k \) converge as \( n o \infty \).
• The convergence may depend on factors like the magnitude and sign of \( a_k \).
• Special techniques, such as log transformations, can be helpful when dealing with products, relating them to the sum convergence.

For instance, the infinite product of cosines leading to \( \frac{\sin x}{x} \) as shown in the exercise implies deep interconnectivity between trigonometric functions and infinite sequences.