Problem 97 - Thomas Calculus

Question

In Exercises \(91-98\) , assume that each sequence converges and find its limit. $$ 2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots $$

Step-by-Step Solution

Answer

The limit of the sequence is \( \sqrt{5} \).

Step 1: Recognize the sequence pattern

The sequence given is of the form \( a_n = 2 + \frac{1}{2 + a_{n-1}} \), where \( a_1 = 2 \). This is a recursive sequence, and we need to identify the pattern for finding the limit as \( n \to \infty \).

Step 2: Assume convergence to a limit

If the sequence converges, let the limit be \( L \). As \( n \to \infty \), both \( a_n \) and \( a_{n-1} \) approach \( L \). Thus, the limit \( L \) satisfies the recurrence relation; i.e., \( L = 2 + \frac{1}{2 + L} \).

Step 3: Solve the equation for L

Substitute \( L \) into the recursive relationship: \( L = 2 + \frac{1}{2 + L} \). Multiply both sides by \( 2 + L \) to eliminate the fraction, resulting in \( L(2 + L) = 2(2 + L) + 1 \). Simplify the equation: \( L^2 + 2L = 4 + 2L + 1 \).

Step 4: Simplify to a quadratic equation

Further simplify to obtain \( L^2 + 2L = 5 + 2L \). Cancel \( 2L \) from both sides: \( L^2 = 5 \). This simplifies to \( L^2 = 5 \).

Step 5: Solve the quadratic equation

Solve \( L^2 = 5 \) to find \( L = \sqrt{5} \) or \( L = -\sqrt{5} \). Since limits typically represent real, positive quantities in this context, the solution is \( L = \sqrt{5} \).

Step 6: Confirm the validity of the solution

Since the sequence is constructed with its elements always being positive, and each term in the sequence, being the sum of 2 and some positive fraction, also reinforces choosing \( L = \sqrt{5} \) rather than \( -\sqrt{5} \).

Key Concepts

Recursive Sequence

A recursive sequence is a sequence of numbers where each term after the first couple of terms is defined as a function of the previous terms. This essentially means you need one or more initial terms and a rule to obtain further terms from the initial ones. Such sequences inherently depend on their prior elements, which you can think of as a "chain" connecting each term to its predecessors.

This type of sequence may seem complex due to its reliance on previous values, but it allows for the generation of intricate series of numbers with relatively simple formulas. This can be especially useful in various applications, such as computer algorithms, where recursion is a fundamental principle.

• Each term in the sequence is linked to the term before it through a specific formula.
• The sequence in the exercise is recursive because each term is calculated by adding a fraction involving the previous term.

Understanding recursive sequences is key in mathematics, as they help develop problem-solving skills by teaching you how to predict and calculate future sequence terms based on a given rule.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence tend toward as the number of terms goes to infinity. Finding the limit is an essential task in calculus since it helps to understand the behavior of sequences and whether they are converging to a particular number.

For the sequence to converge, the terms must consistently get closer to a certain finite number, no matter how large the sequence gets. In other words, the farther you progress in the sequence, the closer you get to the limit.

• If you say a sequence has a limit, it means that as you move along the sequence to very high terms, the difference between the sequence terms and the limit shrinks to zero.
• In specific terms, the limit exists only if, given any small distance (no matter how tiny), there is a point in the sequence after which all subsequent terms of the sequence are within this distance from the limit.

In our exercise example, the sequence converges to a specific limit, which is calculated using both recursion and algebra to find \( L = \sqrt{5} \).

Quadratic Equation

A quadratic equation is an equation where the highest exponent of the variable is a square. It is typically written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Finding solutions (roots) to quadratic equations is a common and important task in algebra.

In solving a quadratic equation, tools such as the quadratic formula, factoring, or completing the square can be employed. The solutions can be real or complex numbers, depending on the discriminant \( b^2 - 4ac \).

• Our step-by-step solution was simplified to the quadratic equation \( L^2 = 5 \), which is a straightforward form of a quadratic equation.
• The roots were found directly by solving this equation, yielding \( L = \sqrt{5} \) and \( L = -\sqrt{5} \).

This illustrates how even nested and recursive sequences can boil down to solving quadratic equations, a foundational skill in algebra.

Convergence Analysis

Convergence analysis involves inspecting whether a sequence or a series tend toward a particular limit as the number of terms continues to increase. It’s essential in various branches of mathematics, as it provides insights into the stability and behavior of sequences over time.

In convergence analysis, one checks whether differences between sequence terms shrink toward zero, effectively determining if a sequence approaches a particular limit. Various tests and criteria exist for systematically conducting this analysis.

• Convergence ensures predictability and consistency within sequences, critical in both theoretical mathematics and applied contexts like engineering and computer science.
• The analysis of convergence in our example involves assuming a limit, formulating it into an equation, and solving for a valid limiting value \( L \).

In practical terms, convergence analysis helps conclude about sequence behavior, as we found that our recursive sequence converges to \( L = \sqrt{5} \), meaning every term inclines toward this numeric value as the sequence progresses.