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## 数学代写|数学物理方法代写Mathematical Methods代考|Bounded, unbounded, convergent, oscillatory

Bounded, unbounded, convergent, oscillatory. Let $M$ be an arbitrary positive number; it is possible that whatever $M$ we take there is at least one value of $8_{n}$ such that $\left|s_{n}\right|>M$. Such a sequence is called unbounded. $s_{n}=n$ is an obvious example, for we need only take $n$ to be any integer greater than $M$. By an argument similar to that for limit-points, an unbounded sequence must have an infinite number of terms such that $\left|s_{n}\right|$ is greater than any assigned $M$.

If we can choose an $M$ such that all $\left|s_{n}\right|$ are less than $M$, the sequence is called bounded. Both the sequences given at the end of $1.04$ are bounded; the condition holds for both if $M=3$.

If there is a number $s$ such that, given any positive number $\epsilon$, we can choose $m$ so that for every $n>m \quad\left|s_{n}-8\right|<\epsilon$, (1) the sequence is said to be convergent, and to have limit 8 . We then write* or $$\begin{gathered} 8_{n} \rightarrow 8 \quad(n \rightarrow \infty) \ \lim {n \rightarrow \infty} 8{n}=8 . \end{gathered}$$ The arrow is read ‘tends to’. We can write simply $$\lim s_{n}=8,$$ if no ambiguity is possible. Of the above examples $1 \cdot 04$ (1) is convergent with limit 0 ; we need only take $m>1 / \epsilon .1 \cdot 04(2)$ is not, because whatever $s$ and $m$ we take, if $\epsilon<\frac{1}{2}$, there will be terms with $n>m$ such that $\left|s_{n}-8\right| \geqslant \frac{1}{2}>\epsilon$.

The most important property of a convergent sequence is that if we have a rule for calculating each term, then we can calculate the limit to any accuracy we like. Some methods of approximation (cf. Chapters 9,17 ) will prove that a quantity lies within a given range, but this range is not arbitrarily small; the accuracy may be enough for the application in view but is not capable of being improved indefinitely.

A sequence that is bounded but not convergent is said to oscillate finitely, or simply to oscillate. An example is $1 \cdot 04(2)$; another is
$$s_{n}=(-1)^{n}+\frac{1}{n} .$$
Unlike 1.04 (2), all $s_{n}$ are different. The sequence is bounded, because $\left|s_{n}\right|<2$ for every $n$; but it does not converge since for large $n$ the members are alternately neer to 1 and $-1$, and (1) cannot be satisfied if $\epsilon<\frac{1}{2}$. If for any $M$ there is an $m$ such that $s_{n}>M$ for all $n>m$, we write
$s_{n}=n$ and $s_{n}=n^{2}$ are examples.
$$s_{n} \rightarrow \infty \text {. }$$
If for any $M$ there is an $m$ such that $s_{n}<-M$ for all $n>m$, we write
$s_{n}=-n$ and $s_{n}=-n^{2}$ are examples.
$$s_{n} \rightarrow-\infty \text {. }$$
Other types of unbounded sequences are represented by
$$s_{n}=(-1)^{n} n, \quad s_{n}=n \cos \frac{1}{2} \pi n, \quad s_{n}=n(1-\cos \pi n) .$$

## 数学代写|数学物理方法代写Mathematical Methods代考|Upper and lower bounds

Upper and lower bounds. A set (or sequence) bounded above has an upper bound; and one bounded below has a lower bound. The upper bound of a set is a quantity $M$ such that no member of the set exceeds $M$, but if $\epsilon$ is any positive quantity, however small, there is a member that exceeds $M-\epsilon$. The lower bound is a quantity $m$ such that no member is less than $m$, but there is always one less than $m+\epsilon$.

We use the method of Dedekind section. There are quantities $a$ such that $a$ is exceeded by some member of the set; for we might take an $a$ less than a known member of the set. Since the set is bounded above, there are quantities $b$ that are not exceeded by any member of the set. Every $b$ is greater than any $a$, and every quantity of the same dimensions is either an $a$ or a $b$. Hence the quantities $a$ form an $L$ and $b$ an $R$ class, and determine a cut, say at $M$. $M$ is a member of the $R$ class. For if it was a member of the $L$ class it would be exceeded by some member of the set, say $K$, and there would be no quantities $b$ between $M$ and $K$; hence $M$ would not be the quantity given by the cut. Hence no member of the set exceeds $M$. Also $M-\epsilon$ is in the $L$ class and therefore is exceeded by some member of the set. The corresponding result for lower bounds follows similarly.

The argument does not suppose the set infinite; but for a finite set the greatest of the set is the upper bound. For an infinite set all members may be less than the upper bound; for the set $1 \cdot 04$ (1) the upper bound is 1 and is equal to the first term, but the lower bound is 0 and no actual member is 0 .

What we call the upper bound is often called the least upper bound; and any quantity such that no member of the set exceeds it is then called an upper bound.

# 数学物理方法代写

## 数学代写|数学物理方法代写Mathematical Methods代想| Bounded, unbounded, convergent, oscillatory

(参见第9，17章) 将证明一个量位于给定的范围内，但这个范围不是任意小的;对于所查看的应用程序来说，准确性可能就足够

$$s_{n}=(-1)^{n}+\frac{1}{n} .$$

$$s_{n} \rightarrow \infty$$

$$s_{n} \rightarrow-\infty \text {. }$$

$$s_{n}=(-1)^{n} n, \quad s_{n}=n \cos \frac{1}{2} \pi n, \quad s_{n}=n(1-\cos \pi n)$$

## 数学代写|数学物理方法代写Mathematical Methods代想| Upper and lower bounds

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。