Problem 98 - Thomas Calculus

Question

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ \begin{array}{l}{\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}} \\\ {\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}, \ldots}\end{array} $$

Step-by-Step Solution

Answer

The limit of the sequence is \( \frac{1 + \sqrt{5}}{2} \).

Step 1: Identify the Sequence

Observe that the sequence is given as a nested radical: \(a_1 = \sqrt{1}\), \(a_2 = \sqrt{1 + a_1}\), \(a_3 = \sqrt{1 + a_2}\), and so on. Each term is justified by the previous term in the sequence.

Step 2: Assume the Limit Exists

Assuming the sequence \(a_n\) converges, let's denote the limit by \(L\). Therefore, the sequence satisfies \( a_n \to L \) as \( n \to \infty \).

Step 3: Formulate the Equation

Since \(a_n \) converges to \( L \), it follows from the sequence definition that \( L = \sqrt{1 + L} \).

Step 4: Solve the Equation

To solve \( L = \sqrt{1 + L} \), square both sides to eliminate the square root: \( L^2 = 1 + L \). Rearrange the terms to form a quadratic equation: \( L^2 - L - 1 = 0 \).

Step 5: Apply the Quadratic Formula

Use the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a=1, b=-1, c=-1\). This gives \( L = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \).

Step 6: Select the Appropriate Solution

The only meaningful solution in this context is \( L = \frac{1 + \sqrt{5}}{2} \), since the sequence terms are positive and increasingly larger.

Key Concepts

Convergence of Sequences

Have you ever noticed how some patterns settle down to a single value as they grow? That's precisely what's happening when we talk about the convergence of sequences. In mathematics, a sequence is simply an ordered list of numbers. As we keep listing these numbers, sometimes they approach what is called a "limit." Imagine getting closer and closer to a destination without ever stepping back.
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A sequence converges if, as we move through the sequence, the numbers get arbitrarily close to a specific number that we call the limit. Here are some key points to understand:

• The limit is the value that the terms of the sequence "aim" towards as the sequence progresses.
• It is assumed that the sequence approaches the limit smoothly without any jumps or interruptions.
• Exploring the limit often involves rigorous proof to ensure that it remains consistent within the sequence's context.

This foundational idea underpins the study of infinite sequences, allowing mathematicians to describe behavior over endless terms with clarity and precision.

Nested Radicals

Nested radicals, sometimes called repeated radicals, look complicated but can carry a lot of meaningful structure. Imagine having a situation where a square root is inside another square root repeatedly. That's a nested radical. For example, \( \sqrt{1}, \sqrt{1 + \sqrt{1}}, \sqrt{1 + \sqrt{1 + \sqrt{1}}}, \dots \)
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Understanding them involves recognizing a chain-like dependency between each term: the next term includes the previous one inside a square root. Here are some things to consider:

• Start by understanding the first few terms because they lay the groundwork for the entire sequence.
• Each subsequent term is created by inserting the previous term into the square root, meaning new levels of complexity arise as more terms are added.
• Nesting often leads to sequences that are interesting in nature because they can converge to a finite limit even though they may appear infinite.

In essence, expressions with nested radicals reveal patterns that harmonize simple and complex forms, showing how mathematics can simplify what seems endlessly complicated.

Quadratic Equation Solutions

Solving equations is like finding the paths of certain curves. Quadratic equations, which take the form \( ax^2 + bx + c = 0 \), describe a parabola. Fortunately, we have a reliable tool called the quadratic formula to find solutions:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• This formula allows us to quickly calculate the roots of any quadratic equation.
• Depending on the discriminant \( (b^2 - 4ac) \), solutions might be real and distinct, real and repeated, or complex.

In our nested radical sequence problem, we derived a quadratic equation \( L^2 - L - 1 = 0 \) from the assumption that our sequence settled to a limit \( L \). Using the quadratic formula, we found the roots as \( \frac{1 \pm \sqrt{5}}{2} \). Choosing only the positive solution \( \frac{1 + \sqrt{5}}{2} \), often known as the golden ratio, aligns with the requirement that our sequence's terms are positive.
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This method demonstrates how a variety of equations, including those emerging from converging sequences, can be approached and solved using consistent mathematical techniques.