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Chapter 3: Problem 10

If \(x_{1}2\), show that \(\left(x_{n}\right)\) is convergent. What is its limit?

Short Answer

Expert verified
The sequence \( \left( x_{n}\right) \) is a convergent sequence with limit \( L \) such that \( L = L \), and is independent of the first two terms \( x_{1} \) and \( x_{2} \).

Step by step solution

01

Show that the sequence is bounded

To show that a sequence is bounded, you need to show that there is a number, M such that all elements of the sequence are smaller than M. First, observe that \(x_{3}=\frac{1}{2} \cdot\left(x_{1}+x_{2}\right)\). From there, all subsequent \(x_{n}\) are averages of two preceding terms, which are themselves constructed from \(x_{1}\) and \(x_{2}\). Thus all \(x_{n}\) are sums and differences divided by powers of 2 of \(x_{1}\) and \(x_{2}\). Consequently, all \(x_{n}\) will fall between the lower and upper bounds given by \(x_{1}\) and \(x_{2}\). So the sequence is bounded.
02

Show that the sequence is Cauchy

A sequence is Cauchy if for any given positive number \( \epsilon \), the terms of the sequence become arbitrarily close to each other as the sequence progresses. We know that \(x_{n}=\frac{1}{2} \cdot\left(x_{n-2}+x_{n-1}\right)\). The difference between two successive terms \(x_{n+1} - x_{n} = \frac{1}{2} \cdot\left(x_{n-1}-x_{n-2}\right)\). As \(n\) gets larger, this difference decreases geometrically and approaches zero, which shows that the sequence is Cauchy.
03

Determine the limit

Given that every term in the sequence other than the first two is the average of its two preceding terms, if there is a limit \(L\), then it must satisfy \(L= \frac{1}{2}(L + L)\). So \(L = L\). The limit does not depend on the initial two terms \(x_{1}\) and \(x_{2}\); it only requires that the sequence is bounded and Cauchy.

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Most popular questions from this chapter

List the first five terms of the following inductively defined sequences. (a) \(x_{1}:=1, \quad x_{n+1}=3 x_{n}+1\), (b) \(y_{1}:=2, \quad y_{n+1}=\frac{1}{2}\left(y_{n}+2 / y_{n}\right)\) (c) \(z_{1}:=1, \quad z_{2}:=2, \quad z_{n+2}:=\left(z_{n+1}+z_{n}\right) /\left(z_{n+1}-z_{n}\right)\) (d) \(s_{1}=3, \quad s_{2}:=5, \quad s_{n+2}:=s_{n}+s_{n+1}\).

Let \(x_{n}:=n^{1 / n}\) for \(n \in \mathbb{N}\). (a) Show that \(x_{n+1}

Establish the convergence and find the limits of the following sequences. (a) \(\left((1+1 / n)^{n+1}\right)\) (b) \(\left((1+1 / n)^{2 n}\right)\) (c) \(\left(\left(1+\frac{1}{n+1}\right)^{n}\right)\), (d) \(\left((1-1 / n)^{n}\right)\)

Suppose that every subsequence of \(X=\left(x_{n}\right)\) has a subsequence that converges to \(0 .\) Show that \(\lim X=0\)

If \(0
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