Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Assuming that \(u_{1}=\sqrt{3}\) and \(u_{n+1}=\sqrt{3+u_{n}}\) determine a convergent sequence, find \(\lim _{n \rightarrow \infty} u_{n}\) to four decimal places.

Step-by-Step Solution

Answer

The sequence converges to 2.3028.

Step 1: Identify the Pattern

Given the sequence defined by the recursive relation, start by calculating the first few terms manually to identify any patterns. Use the formula \( u_{n+1} = \sqrt{3 + u_n} \) with \( u_1 = \sqrt{3} \) to determine \( u_2, u_3, \) and so on.

Step 2: Calculate Initial Terms

Calculate the second term: \( u_2 = \sqrt{3 + u_1} = \sqrt{3 + \sqrt{3}} \). Calculate the third term: \( u_3 = \sqrt{3 + u_2} = \sqrt{3 + \sqrt{3 + \sqrt{3}}} \). This helps to establish values approached by the sequence.

Step 3: Analyze Convergence

Look for convergence behavior. If each term is larger than the previous, but increases at a decreasing rate, consider the possibility of convergence. Visualize or estimate the convergence using numerical approaches if necessary.

Step 4: Solve the Limit Equation

Assume \( u_n \) converges to a limit \( L \). Then \( L = \sqrt{3 + L} \). By solving this equation, you can find the potential limit. Rearrange to solve: set \( L = x \), thus \( x^2 = 3 + x \).

Step 5: Solve the Quadratic Equation

Solve the quadratic equation \( x^2 - x - 3 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1, b = -1, c = -3 \). Calculate the solutions.

Step 6: Determine the Logical Solution

Calculate the roots \( x = \frac{1 \pm \sqrt{13}}{2} \). The two solutions are approximately \( x_1 = 2.3028 \) and \( x_2 = -1.3028 \). Since a sequence of square roots must be non-negative, the valid limit for the sequence is \( x_1 \).

Step 7: Confirm the Limit is Achieved

Substitute or numerically iterate additional terms if needed to ensure the sequence approaches the solution \( x_1 = 2.3028 \) as terms increase.

Key Concepts

Convergence

In calculus, a sequence is said to converge if it approaches a specific value as the number of terms increases indefinitely. This specific value is known as the limit of the sequence. Convergent sequences are key in understanding the behavior of series and functions. To determine convergence:

• Observe the behavior of the sequence: Determine if the terms tend to settle towards a particular value.
• Check if the difference between successive terms decreases over time.
• Use mathematical tools or visual plots to infer and assess the approaching value.

Convergence means stability in long term. When a pattern or regularity emerges, it indicates the sequence is not diverging, i.e., spreading out or moving in an unbounded manner.

Recursive Relations

Recursive relations are equations that express each term in a sequence based on the previous one or more terms. They are like a blueprint for building sequences step-by-step. In the given exercise, the recursive relation is:

• \( u_{n+1} = \sqrt{3 + u_n} \)

This tells us each term is derived by taking the square root of the sum of 3 and the previous term. The initial term, or the starting point, is crucial because it determines the entire sequence's direction. Recursive sequences are often employed to model processes that depend on their previous state, much like population models or financial investments.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence get closer to as the number of terms goes to infinity. We denote the limit of a sequence \( \{u_n\} \) as \( \lim_{n \to \infty} u_n \). In sequences defined by recursive relations, like our exercise, finding the limit involves solving equations where the future terms equal the old ones:

• Set the future term equal to the previous term in its given form: \( L = \sqrt{3 + L} \).
• Solve for \( L \) to find the potential limit that satisfies the condition over the sequence's indefinite span.

Knowing a sequence's limit helps predict long-term behavior of any series modeled by the sequence.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \). Solving these equations can often provide solutions critical to sequences, especially when finding limits involves quadratic forms. The standard method to solve quadratic equations is the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Using our exercise's example, the equation \( x^2 - x - 3 = 0 \) was solved to find: the values for \( L \), the limit of the sequence. The discriminant \( b^2 - 4ac \) tells us about potential solutions. If it's positive, two real solutions exist. In our case:

• Solutions were \( 2.3028 \) and \(-1.3028 \), but only non-negative solutions make sense for this sequence based on square roots.

Understanding quadratics is essential in analyzing the stability and convergence in sequences.