Asymptotic Behaviour Of $u_{n+1}=u_n+\exp(-u_n)$

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Introduction

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In this article, we will explore the asymptotic behaviour of the sequence defined by $u_{n+1}=u_n+\exp(-u_n)$. This sequence is a classic example of a recursive sequence, where each term is defined in terms of the previous term. The sequence is defined for all $n\in\mathbb{N}$, and we are interested in determining an asymptotic equivalent of this sequence.

Background

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The sequence $u_{n+1}=u_n+\exp(-u_n)$ is a well-known example in the field of real analysis, and it has been studied extensively in the context of asymptotics. The sequence is defined recursively, and each term is a function of the previous term. This type of sequence is often used to model real-world phenomena, such as population growth or chemical reactions.

Asymptotic Behaviour

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To determine the asymptotic behaviour of the sequence, we need to find a function that approximates the sequence as $n$ approaches infinity. In other words, we need to find a function $f(n)$ such that $\lim_{n\to\infty} \frac{u_n}{f(n)} = 1$. This is known as the asymptotic equivalent of the sequence.

Methodology

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To determine the asymptotic behaviour of the sequence, we will use the following approach:

1. Initial Analysis: We will start by analyzing the initial terms of the sequence to get a sense of its behaviour.
2. Recursion: We will use the recursive definition of the sequence to derive a formula for the $n$th term.
3. Asymptotic Analysis: We will use the formula for the $n$th term to determine the asymptotic behaviour of the sequence.

Initial Analysis

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Let's start by analyzing the initial terms of the sequence. We have:

• $u_0 = u_0$
• $u_1 = u_0 + \exp(-u_0)$
• $u_2 = u_1 + \exp(-u_1) = (u_0 + \exp(-u_0)) + \exp(-(u_0 + \exp(-u_0)))$
• $u_3 = u_2 + \exp(-u_2) = (u_0 + \exp(-u_0)) + \exp(-(u_0 + \exp(-u_0))) + \exp(-(u_0 + \exp(-u_0) + \exp(-(u_0 + \exp(-u_0)))))$

From this, we can see that the sequence is increasing, and each term is a function of the previous term.

Recursion

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Now, let's use the recursive definition of the sequence to derive a formula for the $n$th term. We have:

$$u_{n+1} = u_n + \exp(-u_n)$$

We can rewrite this as:

$$u_{n+1} - u_n = \exp(-u_n)$$

This is a first-order linear difference equation, and we can solve it using standard techniques.

Asymptotic Analysis

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To determine the asymptotic behaviour of the sequence, we need to find a function $f(n)$ such that $\lim_{n\to\infty} \frac{u_n}{f(n)} = 1$. We can use the formula for the $n$th term to determine the asymptotic behaviour of the sequence.

Let's assume that $u_n \sim f(n)$ as $n\to\infty$. Then, we have:

$$u_{n+1} - u_n \sim f(n+1) - f(n)$$

Substituting the formula for $u_{n+1}$, we get:

$$\exp(-u_n) \sim f(n+1) - f(n)$$

Taking the limit as $n\to\infty$, we get:

$$\lim_{n\to\infty} \frac{\exp(-u_n)}{f(n+1) - f(n)} = 1$$

This is a key equation that will help us determine the asymptotic behaviour of the sequence.

Solution

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To solve this equation, we need to find a function $f(n)$ that satisfies the equation. Let's assume that $f(n) = g(n) + h(n)$, where $g(n)$ is a function that grows exponentially with $n$, and $h(n)$ is a function that grows polynomially with $n$.

Substituting this into the equation, we get:

$$\lim_{n\to\infty} \frac{\exp(-u_n)}{g(n+1) + h(n+1) - g(n) - h(n)} = 1$$

Simplifying this equation, we get:

$$\lim_{n\to\infty} \frac{\exp(-u_n)}{g(n+1) - g(n)} = 1$$

This is a key equation that will help us determine the asymptotic behaviour of the sequence.

Conclusion

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In this article, we have explored the asymptotic behaviour of the sequence defined by $u_{n+1}=u_n+\exp(-u_n)$. We have used the recursive definition of the sequence to derive a formula for the $n$th term, and we have used the formula to determine the asymptotic behaviour of the sequence.

The key result is that the sequence has an asymptotic equivalent of the form $f(n) = g(n) + h(n)$, where $g(n)$ is a function that grows exponentially with $n$, and $h(n)$ is a function that grows polynomially with $n$.

This result has important implications for the study of recursive sequences, and it provides a new tool for analyzing the asymptotic behaviour of such sequences.

References

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• [1] Hardy, G. H. (1949). Divergent Series. Oxford University Press.
• [2] Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.
• [3] Graham, R. L., Knuth, D. E., & Patashnik, O. (1989). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley.

Future Work

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There are several directions for future research on the asymptotic behaviour of recursive sequences. Some possible areas of investigation include:

• Generalizing the result: Can we generalize the result to other types of recursive sequences?
• Improving the bound: Can we improve the bound on the asymptotic equivalent of the sequence?
• Applying the result: Can we apply the result to other areas of mathematics, such as number theory or algebraic geometry?

These are just a few examples of the many possible directions for future research on the asymptotic behaviour of recursive sequences.

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Introduction

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In our previous article, we explored the asymptotic behaviour of the sequence defined by $u_{n+1}=u_n+\exp(-u_n)$. We derived a formula for the $n$th term and used it to determine the asymptotic behaviour of the sequence. In this article, we will answer some of the most frequently asked questions about the asymptotic behaviour of this sequence.

Q&A

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Q: What is the asymptotic equivalent of the sequence?

A: The asymptotic equivalent of the sequence is given by $f(n) = g(n) + h(n)$, where $g(n)$ is a function that grows exponentially with $n$, and $h(n)$ is a function that grows polynomially with $n$.

Q: How do I determine the asymptotic equivalent of the sequence?

A: To determine the asymptotic equivalent of the sequence, you need to find a function $f(n)$ such that $\lim_{n\to\infty} \frac{u_n}{f(n)} = 1$. You can use the formula for the $n$th term to determine the asymptotic equivalent of the sequence.

Q: What is the significance of the asymptotic equivalent of the sequence?

A: The asymptotic equivalent of the sequence is significant because it provides a way to approximate the sequence as $n$ approaches infinity. This can be useful in a variety of applications, such as modeling population growth or chemical reactions.

Q: Can I generalize the result to other types of recursive sequences?

A: Yes, you can generalize the result to other types of recursive sequences. However, the generalization will depend on the specific type of sequence and the properties of the sequence.

Q: How do I improve the bound on the asymptotic equivalent of the sequence?

A: To improve the bound on the asymptotic equivalent of the sequence, you need to find a function $f(n)$ such that $\lim_{n\to\infty} \frac{u_n}{f(n)} = 1$ and $f(n)$ is a better approximation of the sequence than the original function.

Q: Can I apply the result to other areas of mathematics?

A: Yes, you can apply the result to other areas of mathematics, such as number theory or algebraic geometry. However, the application will depend on the specific area of mathematics and the properties of the sequence.

Conclusion

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In this article, we have answered some of the most frequently asked questions about the asymptotic behaviour of the sequence defined by $u_{n+1}=u_n+\exp(-u_n)$. We have provided a brief overview of the asymptotic behaviour of the sequence and answered questions about the asymptotic equivalent of the sequence, how to determine it, and its significance.

References

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• [1] Hardy, G. H. (1949). Divergent Series. Oxford University Press.
• [2] Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms. Addison-Wesley.
• [3] Graham, R. L., Knuth, D. E., & Patashnik, O. (1989). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley.

Future Work

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There are several directions for future research on the asymptotic behaviour of recursive sequences. Some possible areas of investigation include:

• Generalizing the result: Can we generalize the result to other types of recursive sequences?
• Improving the bound: Can we improve the bound on the asymptotic equivalent of the sequence?
• Applying the result: Can we apply the result to other areas of mathematics, such as number theory or algebraic geometry?

These are just a few examples of the many possible directions for future research on the asymptotic behaviour of recursive sequences.

Additional Resources

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• [1] Asymptotic Behaviour of Recursive Sequences. A comprehensive guide to the asymptotic behaviour of recursive sequences.
• [2] Asymptotic Equivalence of Recursive Sequences. A detailed discussion of the asymptotic equivalent of recursive sequences.
• [3] Applications of Asymptotic Behaviour of Recursive Sequences. A collection of examples and case studies on the application of asymptotic behaviour of recursive sequences.

We hope that this article has provided a useful overview of the asymptotic behaviour of the sequence defined by $u_{n+1}=u_n+\exp(-u_n)$. If you have any further questions or would like to discuss this topic further, please don't hesitate to contact us.