text: Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法
  
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  数学代写|数学物理方法代写Mathematical Methods代考|Theorem ofbounded convergence
  
  Theorem of bounded convergence. Uniformity of convergence is a sufficient condition for continuity or integrability of the sum, provided the separate terms are continuous or integrable. It is far from a necessary condition. In practice it is usually easier to test directly whether the limit function is integrable than to test for uniform convergence, and there are so many cases where the passage to the limit under the integral sign is valid without convergence being uniform that a more general rule is needed. Such a rule is as follows. It is known as the theorem of bounded convergencea. If for all $a \leqslant x \leqslant b,\left|f_n(x)\right| \leqslant M$ for all $n$ and $x$, if all $f_n(x)$ are integrable and if $f_n(x) \rightarrow f(x)$, where $f(x)$ is integrable, then $\int_a^b f_n(x) d x \rightarrow \int_a^b f(x) d x$. The proof is not easy, but the result should be known. The behaviour of $f_n(x)$ and
  $$\lim {n \rightarrow \infty} \int_0^1 f_n(x) d x, \int_0^1 \lim {n \rightarrow \infty} f_n(x) d x$$
  should be studied for the cases
  $$f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x}$$
  
  数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
  
  1.117. Useful comparison series. By far the most important comparison series are $\Sigma x^n(0 \leqslant x<1), \Sigma n^{-s}(s>1)$, which we have already studied, and $\Sigma n^s a^n(0 \leqslant a<1)$. The convergence of the latter follows at once from the $M$ test if $s<0$. If $s \geqslant 0$, we have $$\frac{u_{n+1}}{u_n}=\left(\frac{n+1}{n}\right)^s a=\left(1+\frac{1}{n}\right)^{\circ} a \text {. }$$ As $n$ increases this tends to $a$. Hence, since $a<1$, we can take $m$ large enough for this ratio to be less than $b$ for all $n>m$, where $am$,
  $$\left|u_n\right|<u_m b^{n-m}$$
  and $\Sigma b^{n-m}$ is a convergent series of positive terms. Hence $\Sigma n^s a^n$ converges for $0 \leqslant a<1$. Comparison with the series $\Sigma n^{-s}$ can often be simplified. If $v_n=n^{-s}(1<8)$,
  $$n\left(1-\frac{v_n}{v_{n-1}}\right)=n\left{1-\left(\frac{n-1}{n}\right)^s\right} \rightarrow 8$$
  
  数学物理方法代写
  
  数学代写|数学物理方法代写Mathematical Methods代考|Theorem ofbounded convergence
  
  $$\lim n \rightarrow \infty \int_0^1 f_n(x) d x, \int_0^1 \lim n \rightarrow \infty f_n(x) d x$$
  
  $$f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x}$$
  
  数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
  
  1.117。有用的比较系列。到目前为止，最重要的比较系列是 $\Sigma x^n(0 \leqslant x<1), \Sigma n^{-s}(s>1)$ ，我们已经研究 过，并且 $\Sigma n^s a^n(0 \leqslant a<1)$. 后者的收玫立即来自 $M$ 测试是否 $s<0$. 如果 $s \geqslant 0$ ，我们有 $$\frac{u_{n+1}}{u_n}=\left(\frac{n+1}{n}\right)^s a=\left(1+\frac{1}{n}\right)^{\circ} a .$$ 作为 $n$ 增加这往往 $a$. 因此，因为 $a<1$ ，我们可以采取 $m$ 大到足以使这个比率小于 $b$ 对全部 $n>m$ ，在哪里 $a m$
  $$\left|u_n\right|<u_m b^{n-m}$$
  
  \left 缺少或无法识别的分隔符
  
  avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。
  
  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。
explanation: The provided text is an excerpt from a mathematics textbook or lecture notes, specifically dealing with advanced topics in mathematical methods, such as the theorem of bounded convergence and useful comparison series. The text contains a detailed explanation of the theorem of bounded convergence, including conditions for the theorem to hold and an example to study. It also introduces useful comparison series and provides a proof for the convergence of a specific series.
  
  The text does not contain a direct question or a question-and-answer format. It is purely expository, providing definitions, theorems, and proofs.
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: Posted on Categories:Mathematical Methods, 数学代写, 数学物理方法
  
  avatest™帮您通过考试
  
  avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！
  
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  •200+ 英语母语导师
  
  •70分以下全额退款
  
  数学代写|数学物理方法代写Mathematical Methods代考|Theorem ofbounded convergence
  
  Theorem of bounded convergence. Uniformity of convergence is a sufficient condition for continuity or integrability of the sum, provided the separate terms are continuous or integrable. It is far from a necessary condition. In practice it is usually easier to test directly whether the limit function is integrable than to test for uniform convergence, and there are so many cases where the passage to the limit under the integral sign is valid without convergence being uniform that a more general rule is needed. Such a rule is as follows. It is known as the theorem of bounded convergencea. If for all $a \leqslant x \leqslant b,\left|f_n(x)\right| \leqslant M$ for all $n$ and $x$, if all $f_n(x)$ are integrable and if $f_n(x) \rightarrow f(x)$, where $f(x)$ is integrable, then $\int_a^b f_n(x) d x \rightarrow \int_a^b f(x) d x$. The proof is not easy, but the result should be known. The behaviour of $f_n(x)$ and
  $$\lim {n \rightarrow \infty} \int_0^1 f_n(x) d x, \int_0^1 \lim {n \rightarrow \infty} f_n(x) d x$$
  should be studied for the cases
  $$f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x}$$
  
  数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
  
  1.117. Useful comparison series. By far the most important comparison series are $\Sigma x^n(0 \leqslant x<1), \Sigma n^{-s}(s>1)$, which we have already studied, and $\Sigma n^s a^n(0 \leqslant a<1)$. The convergence of the latter follows at once from the $M$ test if $s<0$. If $s \geqslant 0$, we have $$\frac{u_{n+1}}{u_n}=\left(\frac{n+1}{n}\right)^s a=\left(1+\frac{1}{n}\right)^{\circ} a \text {. }$$ As $n$ increases this tends to $a$. Hence, since $a<1$, we can take $m$ large enough for this ratio to be less than $b$ for all $n>m$, where $am$,
  $$\left|u_n\right|<u_m b^{n-m}$$
  and $\Sigma b^{n-m}$ is a convergent series of positive terms. Hence $\Sigma n^s a^n$ converges for $0 \leqslant a<1$. Comparison with the series $\Sigma n^{-s}$ can often be simplified. If $v_n=n^{-s}(1<8)$,
  $$n\left(1-\frac{v_n}{v_{n-1}}\right)=n\left{1-\left(\frac{n-1}{n}\right)^s\right} \rightarrow 8$$
  
  数学物理方法代写
  
  数学代写|数学物理方法代写Mathematical Methods代考|Theorem ofbounded convergence
  
  $$\lim n \rightarrow \infty \int_0^1 f_n(x) d x, \int_0^1 \lim n \rightarrow \infty f_n(x) d x$$
  
  $$f_n(x)=x e^{-n x}, f_n(x)=n x e^{-n x}, f_n(x)=n^2 x e^{-n x}$$
  
  数学代写|数学物理方法代写Mathematical Methods代考|Useful comparison series
  
  1.117。有用的比较系列。到目前为止，最重要的比较系列是 $\Sigma x^n(0 \leqslant x<1), \Sigma n^{-s}(s>1)$ ，我们已经研究 过，并且 $\Sigma n^s a^n(0 \leqslant a<1)$. 后者的收玫立即来自 $M$ 测试是否 $s<0$. 如果 $s \geqslant 0$ ，我们有 $$\frac{u_{n+1}}{u_n}=\left(\frac{n+1}{n}\right)^s a=\left(1+\frac{1}{n}\right)^{\circ} a .$$ 作为 $n$ 增加这往往 $a$. 因此，因为 $a<1$ ，我们可以采取 $m$ 大到足以使这个比率小于 $b$ 对全部 $n>m$ ，在哪里 $a m$
  $$\left|u_n\right|<u_m b^{n-m}$$
  
  \left 缺少或无法识别的分隔符
  
  avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。
  
  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。
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from a mathematics textbook or lecture notes, specifically dealing with advanced topics in mathematical methods, such as the theorem of bounded convergence and useful comparison series. The text contains a detailed explanation of the theorem of bounded convergence, including conditions for the theorem to hold and an example to study. It also introduces useful comparison series and provides a proof for the convergence of a specific series.
  
  The text does not contain a direct question or a question-and-answer format. It is purely expository, providing definitions, theorems, and proofs.
  
  Label: no-question