Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Approximations to \(\pi\) may be obtained from the sequence $$ x_{1}=3, \quad x_{k+1}=x_{k}-\tan x_{k} . $$ Use the TAN key for tan. (a) Find the first five terms of this sequence. (b) What happens to the terms of the sequence when \(x_{1}=6 ?\)

Step-by-Step Solution

Answer

Part (a): The first five terms with \( x_1=3 \) approximate to \( 3, 2.1425, 1.5271, 1.2655, 1.1830 \). Part (b): With \( x_1=6 \), the sequence terms diverge.

Step 1: Initialize the sequence

For part (a), begin the sequence with the given initial value, where \( x_1 = 3 \).

Step 2: Calculate the second term

Using the given recursive formula \( x_{k+1} = x_k - \tan(x_k) \), compute \( x_2 \) by evaluating \( x_2 = x_1 - \tan(x_1) = 3 - \tan(3) \). Calculate the value numerically to obtain \( x_2 \).

Step 3: Calculate the third term

Apply the formula again for \( x_3 \), using the result from Step 2: \( x_3 = x_2 - \tan(x_2) \). Calculate this numerically to find \( x_3 \).

Step 4: Calculate the fourth term

Continuing with the recursive relation, find \( x_4 = x_3 - \tan(x_3) \). Compute this to obtain the value of \( x_4 \).

Step 5: Calculate the fifth term

Use the same process for \( x_5 \) by evaluating \( x_5 = x_4 - \tan(x_4) \) and calculate it numerically to find \( x_5 \).

Step 6: Initial Sequence Setup with x1=6

For part (b), initialize the sequence with \( x_1 = 6 \) as the starting value.

Step 7: Compute Subsequent Terms for x1=6

Calculate subsequent terms \( x_2, x_3, \ldots \) using the same recursive formula. Observe the pattern or behavior of the sequence to understand what happens as more terms are calculated.

Key Concepts

Recursive Sequence

A recursive sequence is a sequence in which terms are related to previous terms through a specific formula or rule. Each term is generated based on the ones that precede it, making it a powerful method for constructing numbers or functions with complex dependencies.
A simple example of a recursive sequence is the Fibonacci sequence, where each number is the sum of the two preceding ones. In the exercise, we use a particular recursive formula: \( x_{k+1} = x_k - \tan(x_k) \).
This means each new term in the sequence is calculated by subtracting from the previous term the tangent of that term. This kind of sequence can quickly become complex, but it is commonly used in mathematical approximations and simulations.

• The sequence starts with an initial value. For instance, starting at \( x_1 = 3 \).
• The recursive formula simplifies the process of calculating further terms by repetition of a single step.
• It allows us to explore how sequences behave as they grow, providing insight into convergence or divergence.

Tangent Function

The tangent function, denoted as \( \tan(x) \), is a trigonometric function that arises in various mathematical contexts. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side. In terms of a unit circle, it is the sine of the angle divided by the cosine. In the sequence described, \( \tan(x_k) \) influences how each term is adjusted from the prior one.
Because \( \tan(x) \) has an periodic behavior hitting unknowns at \( \frac{\pi}{2} + n\pi \), it makes sequences using this function interesting to analyze.

• Important attributes of \( \tan(x) \) include its periodicity and locations of its vertical asymptotes.
• It affects how quickly sequence terms might converge or diverge.

Understanding how the tangent function influences the sequence provides clues about its long-term behavior and helps us approximate values such as \( \pi \). This is especially crucial for mathematical calculations and problem-solving where such functions are manipulated for exactitudes.

Numerical Approximation

Numerical approximation is a method used to find an approximate value for mathematical expressions or quantities that cannot be easily calculated explicitly. In our sequence problem, we utilize numerical approximation to find values of the terms, particularly those resulting from the tangent function, which do not yield simple, closed-form solutions.
Using numerical methods allows computation of values like \( 3 - \tan(3) \) with the help of calculators or computational software.
The primary goal of numerical approximation is to produce a sufficiently accurate answer where straightforward calculations could be too complex or impossible.

• Enables exploration of mathematical ideas beyond analytical boundaries, such as transcendental numbers like \( \pi \).
• Used heavily in simulations, engineering, and physics where exact values are often impractical.

By approximating terms, we gain an understanding of what the sequence's behavior looks like and how it moves closer or further from a particular value like \( \pi \). This understanding can then be applied to similar mathematical challenges.

Sequence Convergence

Sequence convergence refers to the behavior of a sequence as the number of terms increases and whether it approaches a specific value. For the sequence formed by the recursive formula \( x_{k+1} = x_k - \tan(x_k) \), convergence examines whether repeated application of this process leads to a stable result. In the exercise given, convergence is key to understand if the terms are approaching \( \pi \) or another value.
Convergence is said to occur if, as the number of terms becomes very large, the sequence heads toward a single limit and stays arbitrarily close.

• A convergent sequence stabilizes at a particular point, even if it starts with various initial values.
• Understanding convergence helps in predicting outcomes of iterative processes such as algorithms.

By altering the initial values and observing the resulting behavior of the sequence, students can see practical examples of convergence, as well as conditions under which divergence can occur. This concept is fundamental in calculus and analysis, as it allows mathematicians to predict outcomes based on recurring iterative processes.