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Chapter 9: Problem 47

The sequence whose terms are \(1,1,2,3,5,8,13,21, \ldots\) is called the Fibonacci sequence in honor of the Italian math- ematician Leonardo ( Fibonacci') da Pisa (c. \(1170-1250\) ). This sequence has the property that after starting with two 1's, each term is the sum of the preceding two. (a) Denoting the sequence by \(\left\\{a_{n}\right\\}\) and starting with \(a_{1}=1\) and \(a_{2}=1,\) show that $$ \frac{a_{n+2}}{a_{n+1}}=1+\frac{a_{n}}{a_{n+1}} \quad \text { if } n \geq 1 $$ (b) Give a reasonable informal argument to show that if the sequence \(\left\\{a_{n+1} / a_{n}\right\\}\) converges to some limit \(L\), then the sequence \(\left\\{a_{n+2} / a_{n+1}\right\\}\) must also converge to \(L .\) (c) Assuming that the sequence \(\left\\{a_{n+1} / a_{n}\right\\}\) converges, show that its limit is \((1+\sqrt{5}) / 2\).

Short Answer

Expert verified
(a) The ratio equation holds by substituting the recurrence relation. (b) Both sequences must converge to the same limit for consistency. (c) The limit is \( \frac{1 + \sqrt{5}}{2} \).

Step by step solution

01

Express the Fibonacci Sequence Recurrence

The given Fibonacci sequence follows the recurrence relation where each term is the sum of the two preceding ones. This can be expressed as:\[ a_{n+2} = a_{n+1} + a_n \] where \( a_1 = 1 \) and \( a_2 = 1 \).
02

Prove Given Ratio Equation

We need to show that \( \frac{a_{n+2}}{a_{n+1}} = 1 + \frac{a_{n}}{a_{n+1}} \) for \( n \geq 1 \).From Step 1, dividing both sides of \( a_{n+2} = a_{n+1} + a_n \) by \( a_{n+1} \), we get:\[ \frac{a_{n+2}}{a_{n+1}} = \frac{a_{n+1}}{a_{n+1}} + \frac{a_n}{a_{n+1}} = 1 + \frac{a_n}{a_{n+1}} \]Thus, the equation holds true.
03

Informal Argument for Convergence

Assume the sequence \( \left\{ \frac{a_{n+1}}{a_n} \right\} \) converges to some limit \( L \). Then, as \( n \to \infty \), \( \frac{a_n}{a_{n+1}} \to \frac{1}{L} \).From Step 2, \( \frac{a_{n+2}}{a_{n+1}} = 1 + \frac{a_n}{a_{n+1}} \) implies \( \frac{a_{n+2}}{a_{n+1}} \) should converge to \( 1 + \frac{1}{L} \). But \( \frac{a_{n+2}}{a_{n+1}} \) and \( \frac{a_{n+1}}{a_n} \) are successive ratios in the sequence, so for consistency, they both must converge to the same limit \( L \).
04

Find the Limit L

To find \( L \), use the relation \( L = 1 + \frac{1}{L} \). Rearranging gives:\[ L^2 = L + 1 \]Solving this quadratic equation, we get:\[ L^2 - L - 1 = 0 \]Using the quadratic formula, \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -1 \), we find:\[ L = \frac{1 \pm \sqrt{5}}{2} \]Since \( L \) is a positive ratio, we select \( L = \frac{1 + \sqrt{5}}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recurrence Relation
In mathematics, a recurrence relation is an equation that defines a sequence based on preceding terms. For the Fibonacci Sequence, this means that each number is the sum of the two before it. Formally, it is described by:
  • Starting conditions: \(a_1 = 1\) and \(a_2 = 1\)
  • Recurrence relation: \(a_{n+2} = a_{n+1} + a_n\) for \(n \geq 1\)
This relation allows us to build the sequence indefinitely, where the first few numbers are \(1, 1, 2, 3, 5, 8, 13,\) and so forth. Recurrence relations are fundamental in understanding sequences because they provide a systematic way to calculate the terms.
They are like recipes providing steps to achieve an end result, making complex sequences more manageable. This simple recurrence relationship not only helps us understand how the Fibonacci sequence is constructed but also supports more advanced mathematical concepts.
Convergence
Convergence is a key concept in sequences where if the sequence approaches a specific value as the terms increase to infinity, it is said to converge. In the Fibonacci Sequence, we look at the ratio of successive terms, \(\frac{a_{n+1}}{a_n}\). If this ratio approaches a specific number as \(n\) becomes very large, we can say the sequence of ratios converges.
In the context of this problem:
  • Assume the ratio \(\frac{a_{n+1}}{a_n}\) approaches a limit \(L\).
  • This implies that the next ratio \(\frac{a_{n+2}}{a_{n+1}}\) also has to approach \(L\).
The idea is that as we progress in any sequence which is defined in terms of itself (like the Fibonacci one), these ratios stabilize and head towards a specific value. This relational consistency is essential in understanding the behavior of sequences and how they behave infinitely, making convergence a powerful tool in analysis.
Golden Ratio
The Golden Ratio, denoted by the Greek letter \(\phi\) (phi), is approximately \(1.6180339887\ldots\). It emerges as a limit in the Fibonacci sequence when we examine the ratio of consecutive Fibonacci numbers as the sequence progresses towards infinity. When you find the limiting value of \(\frac{a_{n+1}}{a_n}\), it approaches the Golden Ratio, \(\phi\).
For the Fibonacci numbers, solving for the limit \(L\) in the recurrence relation yields the equation:
  • \(L = 1 + \frac{1}{L}\)
By rearranging and solving the quadratic, \(L^2 = L + 1\), we find:
  • \(L = \frac{1 + \sqrt{5}}{2}\)
which is the Golden Ratio. This occurrence in the Fibonacci sequence exemplifies the ratio鈥檚 unique properties. It is found in natural phenomena, art, and architecture, portraying its pervasive beauty and importance. Understanding the Golden Ratio through the Fibonacci sequence gives insight into its universal applicability and aesthetic significance.

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Most popular questions from this chapter

Use the Maclaurin series for \(\sinh x\) and \(\cosh x\) to obtain the first four nonzero terms in the Maclaurin series for tanh \(x .\)

(a) Show that the hypotheses of the integral test are satisfied by the series \(\sum_{k=1}^{\infty} 1 /\left(k^{3}+1\right)\) (b) Use a CAS and the integral test to confirm that the series converges. (c) Construct a table of partial sums for \(n=10,20,\) \(30, \ldots, 100,\) showing at least six decimal places. (d) Based on your table, make a conjecture about the sum of the series to three decimal-place accuracy. (e) Use part (b) of Exercise 36 to check your conjecture.

Suppose that the power series \(\sum c_{k}\left(x-x_{0}\right)^{k}\) has radius of convergence \(R\) and \(p\) is a nonzero constant. What can you say about the radius of convergence of the power series \(\sum p c_{k}\left(x-x_{0}\right)^{k}\) ? Explain your reasoning. [Hint: See Theorem \(9.4 .3 .]\)

Find the radius of convergence and the interval of convergence. $$ \sum_{k=1}^{\infty} \frac{5^{k}}{k^{2}} x^{k} $$

Let \(\sum u_{k}\) be a series and define series \(\sum p_{k}\) and \(\sum q_{k}\) so that $$p_{k}=\left\\{\begin{array}{ll}{u_{k},} & {u_{k}>0} \\ {0,} & {u_{k} \leq 0}\end{array} \quad \text { and } \quad q_{k}=\left\\{\begin{aligned} 0, & u_{k} \geq 0 \\\\-u_{k}, & u_{k}<0 \end{aligned}\right.\right.$$ (a) Show that \(\sum u_{k}\) converges absolutely if and only if \(\sum p_{k}\) and \(\sum q_{k}\) both converge. (b) Show that if one of \(\sum p_{k}\) or \(\sum q_{k}\) converges and theother diverges, then \(\sum u_{k}\) diverges. (c) Show that if \(\sum u_{k}\) converges conditionally, then both $$\sum p_{k} \text { and } \sum q_{k} \text { diverge. }$$
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