Question

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.

Short answer

Question: Given the continued fraction expression $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}},$$ show that it can be built using the recurrence relation $$a_0 = 1, a_{n+1} = 1 + \frac{1}{a_n},$$ find the first five terms of the sequence, estimate the limit of the sequence, determine the exact limit, and determine the value of a continuous fraction with specific conditions on a and b. Answer: The continued fraction expression can be built using the given recurrence relation. The first five terms of the sequence are $1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}$. The estimated limit for the sequence is approximately $1.6$. The exact limit of the sequence is the golden mean, $$\frac{1 + \sqrt{5}}{2}$$. The value of a continuous fraction with specific conditions on a and b is $$L = \frac{a + \sqrt{a^2 + 4ab}}{2}.$$

Step-by-step solution

Connect the Expression with the Recurrence Relation

The given expression is a continued fraction: $$ 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}} $$ We need to show that this expression can be built using the given recurrence relation: $$ a_0 = 1, \quad a_{n+1} = 1 + \frac{1}{a_n}, \quad n = 0, 1, 2, 3, \ldots. $$ We can start by calculating the first few terms of the sequence: $$ a_0 = 1 \\ a_1 = 1 + \frac{1}{a_0} = 1 + \frac{1}{1} = 2 \\ a_2 = 1 + \frac{1}{a_1} = 1 + \frac{1}{2} \\ a_3 = 1 + \frac{1}{a_2} = 1 + \frac{2}{3} \\ a_4 = 1 + \frac{1}{a_3} = 1 + \frac{3}{5} \\ $$ Now, let's rewrite the expression using the terms of the sequence \(\\{a_n\\}\): $$ 1+ \frac{1}{2+\frac{1}{1+\frac{1}{1+\cdots}}} = 1+ \frac{1}{a_1} = a_0 \\ 1+\frac{1}{1+\frac{1}{3+\frac{1}{1+\cdots}}} = 1+ \frac{1}{a_2} = a_1 \\ 1+\frac{1}{1+\frac{1}{1+\frac{1}{5+\cdots}}} = 1+ \frac{1}{a_3} = a_2 \\ $$ Hence, the given expression and the recurrence relation are connected. The value of the expression can be interpreted as \(a_n\) as \(n\) goes to infinity.

First Five Terms of the Sequence

We've already calculated the first four terms of the sequence \(\\{a_n\\}\). Let's calculate the fifth term: $$ a_4 = 1 + \frac{1}{a_3} = 1 + \frac{1}{1 + \frac{2}{3}} = 1 + \frac{1}{\frac{5}{3}} = 1 + \frac{3}{5} = \frac{8}{5}. $$ So, the first five terms of the sequence are: \(1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}\).

Estimate the Limit of the Sequence

One way to estimate the limit of the sequence is to observe its behavior. The sequence seems to be converging, and we can graph the terms to visualize the convergence. By observing the graph or checking convergence tests, we can conclude that the sequence \(\left\\{a_n\right\\}\) converges, and we can estimate the limit to be approximately 1.6.

Determine the Exact Limit

Assuming the limit exists, let's call it L. We can write the equation involving the recurrence relation: $$ L = 1 + \frac{1}{L} $$ Solving the equation for L, we can multiply both sides by L and obtain a quadratic equation: $$ L^2 - L - 1 = 0 $$ Solving this quadratic equation, we obtain two solutions: $$ L = \frac{1 \pm \sqrt{5}}{2} $$ As we are looking for the positive root, the limit of the sequence is: $$ L = \frac{1 + \sqrt{5}}{2} $$ So, the exact limit of the sequence is the golden mean: \(\frac{1 + \sqrt{5}}{2}\)

Determine the Value of a Continuous Fraction with Specific Conditions on a and b

We need to determine the value of the following expression: $$ a + \frac{b}{a + \frac{b}{a+\cdots}} $$ Let's call the value of this expression L, and let's assume that this expression converges as well. We can observe that the pattern of the expression repeats, so we can write: $$ L = a + \frac{b}{L} $$ Multiplying both sides by L and isolating L, we obtain a quadratic equation: $$ L^2 - aL - bL = 0 $$ Solving this quadratic equation, we get the following two solutions: $$ L = \frac{a + \sqrt{a^2 + 4ab}}{2} \quad or \quad L = \frac{a - \sqrt{a^2 + 4ab}}{2} $$ As a and b are positive real numbers, the expression \(a + b\) will always be positive. Therefore, we need to choose the positive solution. So, the value of the expression is: $$ L = \frac{a + \sqrt{a^2 + 4ab}}{2} $$

Key concepts

Recurrence Relations

Recurrence relations are mathematical formulas that express terms in a sequence as a function of preceding terms. They are essential when dealing with infinite sequences or iterative processes, such as continued fractions. In this context, a continued fraction can be expressed through the recurrence relation \(a_{n+1} = 1 + \frac{1}{a_n}\) starting from an initial term \(a_0 = 1\). Here, each term of the sequence is derived from the one before it by applying a set formula.

This concept helps in understanding how a complex fraction can be broken down into simpler components, calculated successively. The recurrence relation provides the mechanism for creating a sequence that mirrors the process of building the continued fraction.

By defining the sequence this way, you can compute each term in sequence order and determine patterns or trends, like approaching a limit or exhibiting convergence behavior.

Sequence Convergence

Sequence convergence refers to the property of a sequence approaching a specific value, known as the limit, as the number of terms goes to infinity. In the context of the continued fraction expressed by the recurrence relation, the sequence \(\{a_n\}\) shows convergence.

To assess convergence, observe the behavior of the sequence's terms: as you calculate more terms, you'll see that the values increasingly stabilize around a certain number. By evaluating terms like \(1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}\), you can graph these and get a visual representation indicating stabilization.

Convergence is valuable since it tells us that despite the infinite nature of the sequence, it is not just growing indefinitely but instead reaching a particular ceiling - its limit. This behavior is essential for various mathematical applications, including numerical series, calculus, and continued fraction approximations.

Golden Ratio

The golden ratio is a special number often denoted by the Greek letter \(\phi\) and is approximately equal to 1.6180339887. It is encountered in various fields, such as art, architecture, and nature, due to its aesthetically pleasing properties.

The golden ratio can also be derived from the sequence discussed in the exercise. Solving the quadratic relation \(L = 1 + \frac{1}{L}\), you obtain a quadratic equation \(L^2 - L - 1 = 0\), leading to the solution of \(L = \frac{1 + \sqrt{5}}{2}\), which is the golden ratio. This proves that the limit of our sequence of continued fractions is precisely the golden ratio.

This number's uniqueness arises from its properties in proportion, design, and natural patterns, highlighting its significance beyond mathematics as a bridge to understanding harmonious structures in the world around us.

Quadratic Equations

Quadratic equations are fundamental expressions of the form \(ax^2 + bx + c = 0\). These equations can be solved to find the values of \(x\) that satisfy them. In the context of this exercise, the quadratic equation arises when setting \(L = 1 + \frac{1}{L}\), leading to \(L^2 - L - 1 = 0\).

Solving this equation by the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) allows you to find the roots that determine the limit of the sequence. Here, the solutions \(L = \frac{1 \pm \sqrt{5}}{2}\) are derived, and since we seek a positive numerical solution, we conclude with the golden ratio.

Quadratic equations are not just abstract mathematical tools; they provide solutions to problems involving parabolic paths, optimization, and, as seen here, the evaluation of continued fractions' limits when they converge to constants like the golden ratio.