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Chapter 8: Problem 51

Consider the following sequences recurrence relations. Using a calculator, make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. $$a_{n+1}=2 a_{n}+1 ; a_{0}=0$$

Short Answer

Expert verified
Answer: The sequence diverges.

Step by step solution

01

Understand the given information

We are given a sequence defined by the recurrence relation \(a_{n+1}=2a_{n}+1\), and an initial condition \(a_{0}=0\). Our goal is to create a table with at least ten terms and determine if the sequence converges or diverges.
02

Calculate the sequence terms

Using the recurrence relation and the initial condition, we will calculate the first ten terms of the sequence. We will use the formula \(a_{n+1}=2a_{n}+1\) and the value of the previous term to find the next term in the sequence. \(a_0 = 0\) \(a_1 = 2a_0 + 1 = 2 \times 0 + 1 = 1\) \(a_2 = 2a_1 + 1 = 2 \times 1 + 1 = 3\) \(a_3 = 2a_2 + 1 = 2 \times 3 + 1 = 7\) \(a_4 = 2a_3 + 1 = 2 \times 7 + 1 = 15\) \(a_5 = 2a_4 + 1 = 2 \times 15 + 1 = 31\) \(a_6 = 2a_5 + 1 = 2 \times 31 + 1 = 63\) \(a_7 = 2a_6 + 1 = 2 \times 63 + 1 = 127\) \(a_8 = 2a_7 + 1 = 2 \times 127 + 1 = 255\) \(a_9 = 2a_8 + 1 = 2 \times 255 + 1 = 511\) \(a_{10} = 2a_9 + 1 = 2 \times 511 + 1 = 1023\)
03

Analyze the sequence

Now that we have calculated the first ten terms of the sequence, let's analyze them to determine if the sequence converges or diverges. The terms of the sequence are as follows: 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023 We can observe that the sequence is increasing and the terms seem to be doubling and then adding 1 at each step. This means the sequence is growing exponentially and does not seem to have a limit. Therefore, we can conclude that the sequence diverges.

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Key Concepts

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

Convergence and Divergence
Convergence and divergence are fundamental concepts in the study of sequences in mathematics. A sequence is said to converge if its terms approach a specific value as the sequence progresses towards infinity. This specific value is known as the limit of the sequence.

On the other hand, when the terms of a sequence do not approach a specific value but instead continue to increase or decrease without bound, or oscillate indefinitely, the sequence is said to diverge. To determine this, one can inspect the behavior of the sequence鈥檚 terms and look for a pattern or apply certain established tests for convergence.

In our example, analyzing the first ten terms reveals a pattern of exponential growth where each term is double the previous term plus one. The terms increase rapidly without settling towards a number, pushing them to grow indefinitely. This particular behavior leads us to conclude that the sequence does not converge to a limit; hence, it diverges.
Recurrence Relation Calculation
A recurrence relation is a way of defining a sequence where each term is formulated based on the previous terms. To calculate the terms in a sequence defined by a recurrence relation, we apply the formula given in the relation successively, starting from the initial conditions provided.

The process of calculating the terms could be easily visualized by creating a table or simply by writing out the terms in order. As observed in the provided exercise, the recurrence relation \(a_{n+1}=2a_{n}+1\) defines each subsequent term as twice the previous term added by 1. Starting from an initial condition \(a_0=0\), the sequence is built by applying this pattern repeatedly.

Example Calculation

  • \(a_1 = 2 \times a_0 + 1\)
  • \(a_2 = 2 \times a_1 + 1\)
And so on. Through iterative calculations like these, the sequence becomes apparent, and we can then study its properties.
Sequence Growth Analysis
Analyzing the growth of a sequence is crucial to understanding its long-term behavior and whether it converges or diverges. The rate of growth can provide insights into the nature of the sequence and how quickly its terms increase or decrease over time.

In examining the growth of a sequence generated by a recurrence relation, we pay attention to the rate at which the terms expand or contract. For our given sequence \(a_{n+1}=2a_{n}+1\), we notice an exponential growth pattern, indicating that each term is progressively larger than the last.

Identifying Exponential Growth

Exponential growth can be identified when a constant is multiplied by each term to get the next, often leading to larger and larger values. This kind of analysis points out that the sequence does not level off, but rather continues to grow without bound, classifying it as a divergent sequence.

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Most popular questions from this chapter

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Suppose an alternating series \(\sum(-1)^{k} a_{k}\) with terms that are non increasing in magnitude, converges to \(S\) and the sum of the first \(n\) terms of the series is \(S_{n} .\) Suppose also that the difference between the magnitudes of consecutive terms decreases with \(k .\) It can be shown that for \(n \geq 1\) \(\left|S-\left(S_{n}+\frac{(-1)^{n+1} a_{n+1}}{2}\right)\right| \leq \frac{1}{2}\left|a_{n+1}-a_{n+2}\right|\) a. Interpret this inequality and explain why it is a better approximation to \(S\) than \(S_{n}\) b. For the following series, determine how many terms of the series are needed to approximate its exact value with an error less than \(10^{-6}\) using both \(S_{n}\) and the method explained in part (a). (i) \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k}\) (ii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{k \ln k}\) (iii) \(\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\sqrt{k}}\)

Assume that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}\) (Exercises 65 and 66 ) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.

Find a formula for the nth term of the sequence of partial sums \(\left\\{S_{n}\right\\} .\) Then evaluate lim \(S_{n}\) to obtain the value of the series or state that the series diverges.\(^{n \rightarrow \infty}\) $$\sum_{k=1}^{\infty}\left(\tan ^{-1}(k+1)-\tan ^{-1} k\right)$$

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd .} \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N\), the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7\), and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?
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