13.4 污名

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  Sigma 符号

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  Introduction
  ::导言

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  Sayber and Tuscany sell ice pops during the summer for pocket money. One particular weekend, they purchased a package of 30 ice pops from the store.
  ::Sayber和Tuscany在夏天出售冰块,以换取零用钱。一个特殊的周末,他们从商店购买了30个冰块。

  

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  Usually they just offer the ice pops for free, one per customer, and accept tips. This time, Tuscany wonders if they would make more money by charging $0.50 per ice pop. At the same time, Sayber wonders if he might be able to increase his tips by encouraging customers to "outbid" each other. The two children decide to each take 15 ice pops and see who makes the most.
  ::通常情况下,他们只免费提供冰块,每个客户一份,并接受小费。 这次,托斯卡尼怀疑他们能否通过每块冰块收费0.50美元来赚更多的钱。 与此同时,赛伯怀疑他是否能够通过鼓励顾客“出价”来增加小费。两个孩子决定每人拿15个冰块,看看谁赚得最多。

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  How can you calculate how much money each of them makes, assuming Sayber gets a $0.10 tip from the 1st customer and is able to convince each successive customer to double the previous person's tip?
  ::假设Sayber从第一客户那里得到0.10美元的小费, 并且能够说服每位连续客户将上一个客户的小费翻一番?

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  Series and Sigma Notation
  ::系列和Sigma 标注

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  Writing the sum of long lists of numbers that have a specific pattern is not very efficient. Summation notation allows you to use the pattern and the number of terms to represent the same sum in a much more concise way.
  ::写入具有特定模式的长数字列表的总和效率不高。 校对符号允许您使用该模式和术语数, 以更简洁的方式代表相同数字 。

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  A series is a sum of a . The Greek capital letter sigma $\begin{align*}\sum_{}^{}{}\end{align*}$  is used for summation notation because it stands for the letter  $\begin{align*}S,\end{align*}$ as in sum.
  ::希腊大写字母 sigma 用于加和符号,因为它代表字母S,与总和一样。

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  Consider the following general sequence, and note that the subscript for each term is an index telling you the term number:
  ::考虑以下一般顺序,并注意每个术语的下标是显示术语数的索引:

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  $$\begin{align*}a_1, a_2, a_3, a_4, \ldots, a_n\end{align*}$$
  ::a1,a2,a3,a4,a4,a1,a2,a3,a4,an

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  When you write the sum of this sequence in a series, it can be represented as a sum of each individual term or abbreviated using a capital sigma.
  ::当您在一系列中写入此序列的总和时,它可以作为每个单个术语的总和表示,或者使用资本西格玛缩写。

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     Summation Notation Using Sigma for the Sum of a Series
  ::使用Sigma 表示系列数总和的计算符号

  $$\begin{align*}a_1+a_2+a_3+a_4+\ldots+a_n=\sum\limits_{i=1}^n a_i\end{align*}$$

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  The three parts of sigma notation that you need to be able to read are the argument, the lower index, and the upper index. The argument, $\begin{align*}a_i\end{align*}$ , tells you what terms are added together. The lower index, $\begin{align*}i=1\end{align*}$ , tells you where to start, and the upper index, 5, tells you where to end. 
  ::您需要读取的污名符号的三个部分是参数、 较低的索引和上层索引。 参数 Ai 表示将哪些术语加在一起。 下层索引 i=1 表示开始位置, 上层索引 5 表示结束位置 。

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  The following video  explains how to find a sum when given in summation/sigma notation: 
  ::以下影片解释当以总和/污名符号表示时如何找到一个总和:

   

   

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  Play, Learn, and Explore with Sigma Notation: 
  ::使用 Sigma 符号玩、学习、探索:

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  Examples
  ::实例

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  Example 1
  ::例1

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  Write out all the terms of the series.
  ::写出系列的所有条款。

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  $$\begin{align*}\sum\limits_{k=4}^8 2k\end{align*}$$
  ::8k=42k

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  Solution:
  ::解决方案 :

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  $$\begin{align*}\sum\limits_{k=4}^8 2k=2\cdot4+2\cdot5+2\cdot6+2\cdot7+2\cdot8\end{align*}$$
  ::8k=42k=24+25+26+27+28

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  Example 2
  ::例2

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  Write the sum in sigma notation: $\begin{align*}2+3+4+5+6+7+8+9+10\end{align*}$ .
  ::以 sigma 符号写入总和: 2+3+4+5+6+7+8+8+9+10。

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  Solution:
  ::解决方案 :

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  $$\begin{align*}2+3+4+5+6+7+8+9+10=\sum\limits_{i=2}^{10} i\end{align*}$$
  ::2+3+4+4+5+6+6+7+7+8+9+9+10=10i=2i

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  Example 3
  ::例3

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  Write the sum in sigma notation:
  ::以西格玛符号写下总和 :

  $$\begin{align*}1+4+9+16+25+\cdots+144\end{align*}$$

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  Solution:
  ::解决方案 :

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  The hardest part when first using sigma representation is determining how each pattern generalizes to the  $\begin{align*}k^{th}\end{align*}$ term. Once you know the $\begin{align*}k^{th}\end{align*}$ term, you know the argument of the sigma. For the sequence creating the series below, $\begin{align*}a_k=k^2\end{align*}$ . Therefore, the argument of the sigma is $\begin{align*}i^2\end{align*}$ .
  ::当第一次使用 sigma 代表符时最难的部分是决定每个模式如何向 kth 术语概括。 一旦您知道 kth 术语, 您就会知道 sigma 的参数。 对于创建下面序列的顺序, ak=k2 , 因此, sigma 的参数是 i2 。

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  $$\begin{align*}1+4+9+16+25+\cdots+144=1^2+2^2+3^2+4^2+\cdots 12^2=\sum\limits_{i=1}^{12} i^2\end{align*}$$
  ::1+4+9+16+25144=12+22+32+42122=12i=1i2

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  Example 4
  ::例4

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  Recall the problem from the Introduction: How can you calculate how much money Sayber and Tuscany each make, assuming Sayber gets a $0.10 tip from the 1st customer and is able to convince each successive customer to double the previous person's tip?
  ::回顾导言中的问题:假设Sayber从第一客户那里得到0.10美元的小费,并且能够说服每一连续客户将上一个客户的小费翻一番,你如何计算出每个Sayber和Tuscany挣的钱?

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  Solution:
  ::解决方案 :

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  Tuscany's income can be expressed as 

  $$\begin{align*}\sum_{n=1}^{10} 0.50 \to 0.50 \cdot 10 = \$5.\end{align*}$$ Sayber's income can be expressed as  $$\begin{align*}\sum_{n=1}^{10}0.10 \cdot 2^n \to 0.10 (2^0) + 0.10 (2^1) +0.10 (2^2)... +0.10 (2^{10}) \to \$102.30.\end{align*}$$
  ::托斯卡纳的收入可以表示为10n=10.5500.5010=5美元。 赛义伯的收入可以表示为10n=10.102n0.10(20)+0.10(21)+0.10(22)...+0.10(210)__10美元。

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  Example 5
  ::例5

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  Write out all the terms of the sigma notation, and then calculate the sum.
  ::写出所有污名符号的术语, 然后计算总和 。

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  $$\begin{align*}\sum\limits_{k=0}^4 3k-1\end{align*}$$
  ::4k=03k- 1

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  Solution:
  ::解决方案 :

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  $$\begin{align*}\sum\limits_{k=0}^4 3k-1&=(3\cdot0-1)+(3\cdot1-1)+(3\cdot2-1)+(3\cdot3-1)+(3\cdot4-1)\\ &=-1+2+5+8+11=25\end{align*}$$
  ::4k=03k-1=(30-1-1)+(31-1-1)+(32-1-1)+(32-1-1)+(33-3)-1)+(34-1)+(34-1)++1+2+5+8+11=25

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  Example 6
  ::例6

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  Represent the following infinite series in summation notation:
  ::以总和符号表示以下无限序列:

  $$\begin{align*}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots\end{align*}$$

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  Solution:
  ::解决方案 :

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  There are an infinite number of terms in the series, so using an infinity symbol in the upper limit of the sigma is appropriate.
  ::序列中有无限的术语, 所以使用无穷的符号 在西格玛的上限是合适的。

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  $$\begin{align*}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots=\sum\limits_{i=1}^\infty \frac{1}{2^i}\end{align*}$$
  ::12+14+18+116121+122+123+124i=112i

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  Example 7
  ::例7

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  Write the sum in sigma notation.
  ::将这笔金额写在Sigma 符号中。

  $$\begin{align*}\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}\end{align*}$$

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  Solution:
  ::解决方案 :

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  $$\begin{align*}\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}=\sum\limits_{i=1}^7 \frac{1}{i^2}\end{align*}$$
  ::112+122+132+142+152+162+172=7i=11i2

   

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  Summary
  ::摘要

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  • Sigma notation  $\begin{align*}\sum_{}^{}{}\end{align*}$   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 也被称为总和符号 , 是一种代表数字总和的方法。 当数字有特定模式或无需缩写而写得太久时, 它特别有用 。
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  • Sigma notation can be defined as follows:  $\begin{align*}a_1+a_2+a_3+a_4+\ldots+a_n=\sum\limits_{i=1}^n a_i.\end{align*}$
    ::污名符号可定义如下:a1+a2+a3+a4+...+an=ni=1ai。

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  Review
  ::回顾

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  For 1-5, write out all the terms of the sigma notation and then calculate the sum.
  ::1 -5, 写出所有污名符号的术语, 然后计算总和 。

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  1. $\begin{align*}\sum\limits_{k=1}^5 2k-3\end{align*}$
  ::1. 5当k=12k-3

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  2. $\begin{align*} \sum\limits_{k=0}^8 2^k\end{align*}$
  ::2. 8k=02k

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  3. $\begin{align*}\sum\limits_{i=1}^4 2 \cdot 3^i\end{align*}$
  ::3. 4i=123i

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  4. $\begin{align*}\sum\limits_{i=1}^{10} 4i-1\end{align*}$
  ::4. 10-i=14i-1

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  5.   $\begin{align*}\sum_{n=1}^{11} 9(4)^{n-1}\end{align*}$
  ::5. 11n=19(4)n-1

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  Represent the following series in summation notation with a lower index of 0:
  ::代表以下序列,加号为0,较低指数为0:

  6. $\begin{align*}1+4+7+10+13+16+19+22\end{align*}$

  7. $\begin{align*} 3+5+7+9+11\end{align*}$

  8. $\begin{align*}8+7+6+5+4+3+2+1\end{align*}$

  9. $\begin{align*}5+6+7+8\end{align*}$

  10. $\begin{align*}3+6+12+24+48+\cdots\end{align*}$

  11. $\begin{align*}10+5+\frac{5}{2}+\frac{5}{4}\end{align*}$

  12. $\begin{align*}4-8+16-32+64\cdots\end{align*}$

  13. $\begin{align*}2\cdot 4\cdot 6\cdot 8\cdot \cdots\end{align*}$

  14. $\begin{align*}\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots\end{align*}$

  15. $\begin{align*}\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\cdots\end{align*}$

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  Review (Answer s)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。