13.9 摘要:序列和系列

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  摘要:序列和系列

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  In this chapter you learned that recursion, how people see patterns, is where each term in a sequence is defined by the term that came before. You saw, too, that terms in a pattern can also be represented as a function of their term number. Moreover, you learned about two special types of patterns called arithmetic sequences and geometric sequences. Sequences have a wide variety of applications in the real world. Additionally, you saw that series are when terms in a sequence are added together.
  ::在本章中,你了解到,循环,人们如何看待模式, 是一个序列中每个术语的顺序是由之前的术语来定义的。你也看到, 模式中的术语也可以以其术语数的函数来表示。 此外,你还了解到了两种特殊模式,即算术序列和几何序列。序列在现实世界中有着各种各样的应用。此外,你还看到,序列是将一个序列中的术语加在一起的。

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  Chapter Summary
  ::章次摘要

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  • A recursively defined pattern or sequence is a sequence with terms that are defined based on the prior term(s) in the sequence.
    ::循环定义的图案或序列是指根据序列中前一术语界定的术语序列。
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  • An explicit pattern or sequence is a sequence with terms that are defined based on the term number.
    ::明确的模式或顺序是按术语编号界定术语的顺序。
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  • Sigma notation  $\begin{array}{r}\underset{}{\overset{}{\sum }}\end{array}$   is also known as summation notation and is a way to represent a sum of numbers. It is especially useful when the numbers have a specific pattern or would take too long to write out without abbreviation.
    ::Sigma 符号 也称为总和符号 , 是代表数字总和的一种方式。 当数字有特定模式, 或没有缩略语而写得过长时, 它特别有用 。
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  • Sigma notation can be defined as follows:  $\begin{array}{r}{a}_1+a_2+a_3+a_4+\dots +a_n=\underset{i=1}{\overset{n}{\sum }}{a}_i.\end{array}$
    ::Sigma 符号可定义如下: a1+a2+a3+a4+...+ani=1nai。
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  • A sequence is a list of numbers separated by commas.
    ::序列是用逗号分隔的数字列表 。
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  • The common pattern in an arithmetic sequence is that the same number is added or subtracted to each number to produce the next number. This is called the common difference.
    ::算术序列中的常见模式是,为产生下一个数字,在每个数字中加上或减去相同数字,这称为共同差数。
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  • The common pattern in a geometric sequence is that the same number is multiplied or divided to each number to produce the next number. This is called the common ratio.
    ::几何序列中的常见模式是,同一数字乘以或除以每个数字,得出下一个数字。这称为共同比率。
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  • An arithmetic series is a sum of numbers whose consecutive terms form an arithmetic sequence.
    ::算术序列是数字的总和,其连续术语构成算术序列。
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  • The general form for an arithmetic series can be represented by  $\begin{array}{r}\underset{i=1}{\overset{n}{\sum }}{a}_i=\frac{n}{2}(2a_1+(n-1)k),\end{array}$ where k is the common difference for the terms in the series.
    ::算术序列的一般格式可以用“i=1nai=n2(2a1+(n-1)k)表示,K是该序列术语的常见差异。
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  • To converge means the sum approaches a specific number.
    ::趋同意味着求和法采用一个具体的数字。
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  • To diverge means the sum does not converge, and so usually goes to positive or negative infinity. It could also mean that the series oscillates infinitely.
    ::差异意味着总和不会趋同,因此通常会变成正或负的无限。 它还可能意味着序列会无穷地振动。
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  • A partial sum of an infinite sum is the sum of all the terms up to a certain point. Considering partial sums can be useful when analyzing infinite sums.
    ::无限总和的一部分是直到某一点的所有条件的总和。在分析无限总和时,考虑部分总和是有用的。
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  • Notation for the sum of a geometric series:  $\begin{array}{r}{a}_1+a_{1}r+a_1{r}^2+\cdots a_1{r}^{n-1}=a_1\left(\frac{1-r^n}{1-r}\right)\end{array}$ .
    ::a1+a1r+a1r2_a1rn_1=a1(1-rn1-r)。

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  Review 
  ::回顾

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  Try the following cumulative review problems to practice the concepts we studied in this chapter. 
  ::尝试下列累积审查问题,以实践我们在本章中研究的概念。