Problem 39

Question

Find $\lim _{n \rightarrow \infty} u_{n}$ of Problem 37 algebraically. Hint: Let $u=\lim _{n \rightarrow \infty} u_{n} .$ Then, since $u_{n+1}=\sqrt{3+u_{n}}, u=\sqrt{3+u}$. Now square both sides and solve for $u$.

Step-by-Step Solution

Verified

Answer

The limit is $ u = \frac{1 + \sqrt{13}}{2} $.

1Step 1: Understanding the Problem

We need to find the limit $ u $ of a sequence $ u_n $ as $ n $ approaches infinity, where $ u_{n+1} = \sqrt{3+u_n} $. The problem provides a condition for $ u $: $ u = \sqrt{3+u} $.

2Step 2: Setting Up the Equation

Start with the equation given by $ u = \sqrt{3+u} $. This equation arises from the definition of the limit of the sequence as $ n $ approaches infinity.

3Step 3: Solving the Equation

To eliminate the square root, square both sides of the equation: $ u^2 = 3 + u $. Now we have a quadratic equation in terms of $ u $.

4Step 4: Rearrange the Quadratic Equation

Rewrite the equation $ u^2 = 3 + u $ as $ u^2 - u - 3 = 0 $. This is a standard quadratic equation.

5Step 5: Solve the Quadratic Using the Quadratic Formula

The quadratic formula is $ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. For $ u^2 - u - 3 = 0 $, $ a = 1 $, $ b = -1 $, and $ c = -3 $. Substitute these into the formula: $ u = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} $.

6Step 6: Calculate the Discriminant

Calculate the discriminant: $ (-1)^2 - 4 \cdot 1 \cdot (-3) = 1 + 12 = 13 $, so $ u = \frac{1 \pm \sqrt{13}}{2} $.

7Step 7: Find the Valid Solution

The solutions to the quadratic are $ u = \frac{1 + \sqrt{13}}{2} $ and $ u = \frac{1 - \sqrt{13}}{2} $. Since the limit represents a real, positive number, and the sequence involves taking square roots (always non-negative), the valid solution is $ u = \frac{1 + \sqrt{13}}{2} $.

8Step 8: Finalize the Solution

The solution to $ \lim_{n \to \infty} u_n $ is $ u = \frac{1 + \sqrt{13}}{2} $. This is the positive solution from the quadratic equation, consistent with the nature of the sequence.

Key Concepts

Recursive SequencesQuadratic EquationSquare RootsDiscriminant in Quadratic Formula

Recursive Sequences

Recursive sequences form a set of values that are generated based on a specific rule which derives the next term from one or more of the previous terms. In our problem, the sequence is defined as $ u_{n+1} = \sqrt{3+u_{n}} $.
This means each term in the sequence depends on the term immediately before it. In general, recursive sequences can have any rule that relates terms to prior terms.
One powerful aspect of recursive sequences is finding patterns or limits as the sequence progresses to infinity.

Recursive sequences require initial values to start the process. This is often called the base or initial term.
Determining the limit involves finding what value, if any, the terms approach as they progress.

Recursive sequences can model many natural and mathematical processes and understanding them requires observing how they evolve over iterations.

Quadratic Equation

Quadratic equations are polynomial equations of the second degree, which means they contain a squared term as their highest power: $ ax^2 + bx + c = 0 $.
In our example, solving the problem involves converting a recursive relationship into a quadratic equation.
After squaring both sides of the equation derived from the limit $ u=\sqrt{3+u} $, we reach $ u^2 - u - 3 = 0 $.

Every quadratic equation can have two solutions, sometimes real and sometimes complex.
Quadratic equations are solvable by different methods, including factoring, completing the square, and the quadratic formula.

Quadratic equations are fundamental in algebra and have applications in physics, engineering, and more, whenever a relationship between diverging lines needs to be understood.

Square Roots

Square roots involve finding a number which, when multiplied by itself, produces the original value. For a number $ x $, its square root is denoted as $ \sqrt{x} $.
In the recursive sequence, the square root function is used directly, which impacts how the sequence progresses: $ u_{n+1} = \sqrt{3+u_{n}} $.

Square roots are always non-negative in the scope of real numbers, which aligns with our requirement for sequences values, as sequence values must be real and positive.
Understanding square roots' behavior is crucial since they convert the quadratic nature hidden within the sequence structure.

This concept simplifies complex values and is crucial in deriving the solution, especially here, where removing the square root transforms the equation into a solvable quadratic form.

Discriminant in Quadratic Formula

The discriminant in a quadratic equation helps determine the nature of the roots of the quadratic. In the quadratic formula $ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, the part $ b^2 - 4ac $ is the discriminant.
It tells us whether the solutions are real, repeated, or complex. In this case, the discriminant is $ 13 $, which is positive, indicating two distinct real solutions.

A positive discriminant means there are two distinct real roots.
A discriminant of zero suggests one repeated real root.
A negative discriminant results in complex roots.

Understanding the discriminant is critical because it informs you about the potential nature and number of the solutions without fully solving the equation.

Problem 39

Problem 40

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