Section: Sigma 符号 | 12.3 西格玛符号

Section outline

Writing the sum of long lists of numbers that have a specific pattern is not very efficient. Summation or sigma notation allows you to use the pattern and the number of terms to represent the same sum in a much more concise way. How can you use sigma notation to represent the following sum?
::写出具有特定模式的长数字列表的总和效率不高。校对或刻度符号允许您使用该模式和术语数来以更简洁的方式代表相同数字。您如何使用刻度符号来代表下个总和 ?

$\begin{aligned} 1 + 4 + 9 + 16 + 25 + \dots + 144 \end{aligned}$

Series in Sigma Notation
::Sigma 符号系列

A series is a sum of a . Summation notation is also known as sigma notation and is a way to represent a series. It is especially useful when the series would take too long to write out without abbreviation. The Greek capital letter sigma, $\begin{aligned} \sum_{}^{} \end{aligned}$ is used for summation notation because it stands for the letter $\begin{aligned} S \end{aligned}$ as in sum.
::希腊大写字母Sigma, __ 用于总和符号, 因为它代表字母 S 的总和。

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。

$\begin{aligned} a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \end{aligned}$
::a1,a2,a3,a4,a5

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。

$\begin{aligned} a_{1} + a_{2} + a_{3} + a_{4} + a_{5} = \sum_{i = 1}^{5} a_{i} \end{aligned}$
::a1+a2+a3+a4+a5+a5i=15ai

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, $\begin{aligned} a_{i} \end{aligned}$ , tells you what terms are added together. The lower index, $\begin{aligned} i = 1 \end{aligned}$ , tells you where to start and the upper index, 5, tells you where to end. You should practice reading and understanding sigma notation because it is used heavily in Calculus.
::您需要读取的污名符号的三个部分是参数、较低的索引和上一级索引。参数 Ai 表示将哪些术语加在一起。下一级索引 i=1 表示开始位置, 上一级索引 5 表示结束位置。您应该练习读取和理解污名符号, 因为该符号在计算时使用很多。

Take the series:
::采取系列 :

$\begin{aligned} \sum_{k = 4}^{8} 2 k \end{aligned}$
::k=482k

Written out, this would be:
::书面形式如下:

$\begin{aligned} \sum_{k = 4}^{8} 2 k = 2 \cdot 4 + 2 \cdot 5 + 2 \cdot 6 + 2 \cdot 7 + 2 \cdot 8 \end{aligned}$
::k=482k=24+25+26+27+28

Examples
::实例

Example 1
::例1

Earlier, you were asked how to write the sum $\begin{aligned} 1 + 4 + 9 + 16 + 25 + \dots + 144 \end{aligned}$ in sigma notation. The hardest part when first using sigma representation is determining how each pattern generalizes to the $\begin{aligned} k^{t h} \end{aligned}$ term. Once you know the $\begin{aligned} k^{t h} \end{aligned}$ term, you know the argument of the sigma. For the sequence creating the series below, $\begin{aligned} a_{k} = k^{2} \end{aligned}$ . Therefore, the argument of the sigma is $\begin{aligned} i^{2} \end{aligned}$ .
::早些时候, 您被问及如何在 sigma 符号中写入 $ 1+4+9+9+16+25+144 。第一次使用 sigma 符号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。对于下面创建序列的序列, Ak=k2 , 因此, sigma 的参数是 i2 。

$\begin{aligned} 1 + 4 + 9 + 16 + 25 + \dots + 144 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + \dots 12^{2} = \sum_{i = 1}^{12} i^{2} \end{aligned}$
::1+4+9+16+25144=12+22+32+42122i=1112i2

Example 2
::例2

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

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

$\begin{aligned} \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \frac{1}{5^{2}} + \frac{1}{6^{2}} + \frac{1}{7^{2}} = \sum_{i = 1}^{7} \frac{1}{i^{2}} \end{aligned}$
::112+122+132+142+152+162+172+172=171i=171i2

Example 3
::例3

Write out all the terms of the sigma notation and then calculate the sum.
::写出所有刻度符号的条件, 然后计算总和。

$\begin{aligned} \sum_{k = 0}^{4} 3 k - 1 \end{aligned}$
::*k=043k-1

\begin{aligned} \sum_{k = 0}^{4} 3 k - 1 & = (3 \cdot 0 - 1) + (3 \cdot 1 - 1) + (3 \cdot 2 - 1) + (3 \cdot 3 - 1) + (3 \cdot 4 - 1) \\ = - 1 + 2 + 5 + 8 + 11 \end{aligned}

::*k=043k-1=(30-1-1)+(31-1-1)+(32-1-1)+(32-1-1)+(33-3-1-1)+(34-1)+(1)+1+2+5+8+11

Example 4
::例4

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

$\begin{aligned} \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \end{aligned}$

There are an infinite number of terms in the series so using an infinity symbol in the upper limit of the sigma is appropriate.
::序列中有无数个术语,因此在“西格玛”的上限使用无穷符号是合适的。

$\begin{aligned} \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = \frac{1}{2^{1}} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \frac{1}{2^{4}} + \dots = \sum_{i = 1}^{\infty} \frac{1}{2^{i}} \end{aligned}$
::12+14+18+116121+122+123+124i=1112i

Example 5
::例5

Is there a way to represent an infinite product? How would you represent the following product?
::是否有代表无限产品的方法? 您如何代表以下产品 ?

$\begin{aligned} 1 \cdot \sin (\frac{360}{3}) \cdot \sin (\frac{360}{4}) \cdot \sin (\frac{360}{5}) \cdot \sin (\frac{360}{6}) \cdot \sin (\frac{360}{7}) \cdot \dots \end{aligned}$
::-=YTET -伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=- 翻译:

Just like summation uses a capital Greek letter for $\begin{aligned} S \end{aligned}$ , product uses a capital Greek letter for $\begin{aligned} P \end{aligned}$ which is the capital form of $\begin{aligned} π \end{aligned}$ .
::就像加起来用希腊字母表示S, 产品用希腊字母表示P, 这是______的资本形式。

$\begin{aligned} 1 \cdot \sin (\frac{360}{2 \cdot 3}) \cdot \sin (\frac{360}{2 \cdot 4}) \cdot \sin (\frac{360}{2 \cdot 5}) \cdot \sin (\frac{360}{2 \cdot 6}) \cdot \sin (\frac{360}{2 \cdot 7}) \cdot \dots = \prod_{i = 3}^{\infty} \sin (\frac{360}{2 \cdot i}) \end{aligned}$
::

This infinite product is the result of starting with a circle of radius 1 and inscribing a regular triangle inside the circle. Then you inscribe a circle inside the triangle and a square inside the new circle. The shapes alternate being inscribed within each other as they are nested inwards: circle, triangle, circle, square, circle, pentagon, ... The question that this calculation starts to answer is whether this process reduces to a number or to zero.
::此无限产品是以圆半径 1 开始, 并在圆内输入一个普通三角形的结果。然后您在三角内输入一个圆, 在新圆内输入一个正方形。当它们嵌入嵌入时, 这些形状会相互嵌入 : 圆、三角、圆、平方、圆、五角、... 这个计算开始回答的问题是这个过程是减为数还是减为零。

Summary

Summation or sigma notation is a concise way to represent the sum of long lists of numbers with a specific pattern.
::汇总或刻度符号是代表具有特定模式的长清单数字之和的简明方式。

Sigma notation consists of three parts: the argument, the lower index, and the upper index.
::污名化包括三个部分:论点、较低指数和较高指数。

The argument indicates the terms being added
::该论点表明所添加的术语

The lower index shows where to start.
::较低的指数显示起点。

The upper index shows where to end.
::上一个指数显示要结束的地方。

Review
::回顾

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。

1. $\begin{aligned} \sum_{k = 1}^{5} 2 k - 3 \end{aligned}$
::1. =152k-3

2. $\begin{aligned} \sum_{k = 0}^{8} 2^{k} \end{aligned}$
::2.k=082k

3. $\begin{aligned} \sum_{i = 1}^{4} 2 \cdot 3^{i} \end{aligned}$
::3.-i=1423-3i

4. $\begin{aligned} \sum_{i = 1}^{10} 4 i - 1 \end{aligned}$
::4. i=1104i-1

5. $\begin{aligned} \sum_{i = 0}^{4} 2 \cdot {(\frac{1}{3})}^{i} \end{aligned}$
::5. i=042(13)i

Represent the following series in summation notation with a lower index of 0.
::以0的较低指数表示以下序列,加号为总和符号。

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

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

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

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

10. $\begin{aligned} 3 + 6 + 12 + 24 + 48 + \dots \end{aligned}$

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

12. $\begin{aligned} 4 - 8 + 16 - 32 + 64 \dots \end{aligned}$

13. $\begin{aligned} 2 + 4 + 6 + 8 + \dots \end{aligned}$

14. $\begin{aligned} \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \dots \end{aligned}$

15. $\begin{aligned} \frac{2}{3} + \frac{2}{9} + \frac{2}{27} + \frac{2}{81} + \dots \end{aligned}$

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Series in Sigma Notation ::Sigma 符号系列

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)