koorio.com
海量文库 文档专家
当前位置:首页 >> 数学 >>

IFY


IFY Maths

Sequences and Series

Sequences and Series Examples of Sequences e.g. 1 e.g. 2

2, 4, 6, 8, . . .
1 1 1 1, , , , . . . 2 3 4

e.g. 3

1, ? 4, 16, ? 64, . . .
A sequence is an ordered list of numbers

The 3 dots are used to show that a sequence continues

Sequences and Series Recurrence Relations Can you predict the next term of the sequence 3, 5, 7, 9, . . . ? 11 Suppose the formula continues by adding 2 to each term.

The formula that generates the sequence is then

u n ?1 ? u n ? 2
where

u n and u n ?1 are terms of the sequence u1 is the 1st term, so u1 ? 3 n ? 1 ? u2 ? u1 ? 2 ? u2 ? 3 ? 2 ? 5 n ? 2 ? u3 ? u2 ? 2 ? u3 ? 5 ? 2 ? 7
etc.

Sequences and Series Recurrence Relations

A formula such as
recurrence relation

un?1 ? un ? 2 is called a

e.g. 1 Give the 1st term and write down a

recurrence relation for the sequence

1, ? 4, 16, ? 64, . . .
Solution: 1st term: Recurremce relation:

u1 ? 1 un?1 ? ?4 un

Other letters may be used instead of u and n, so the formula could, for example, be given as

a k ?1 ? ?4 a k

Sequences and Series Recurrence Relations e.g. 2 Write down the 2nd, 3rd and 4th terms of the sequence given by Solution:

u1 ? 5,

u i ? 1 ? 2u i ? 3

i ?1 i?2 i?3

? ? ?

u 2 ? 2u 1 ? 3 u 2 ? 2(5) ? 3 ? 7 u 3 ? 2u 2 ? 3 u 3 ? 2(7) ? 3 ? 11
u 4 ? 2u 3 ? 3

?
? ?

The sequence is

u 4 ? 2(11) ? 3 ? 19 5, 7, 11, 19, . . .

Sequences and Series Properties of sequences Convergent sequences approach a certain value e.g. 1, 1 1 , 1 3 , 1 7 , 1 15 . . .
2 4 8 16

approaches 2

un

n

Sequences and Series Properties of sequences Convergent sequences approach a certain value e.g. 1, ? 1 , 1 , ? 1 , 1 , . . .
2 4 8 16

approaches 0

un

n

This convergent sequence also oscillates

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.
un

2, 4, 6, 8, 10, . . .

n

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.

?1, 2, ? 4, 8, ? 16, . . .
un

n

This divergent sequence also oscillates

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.

1, 2, 3, 1, 2, 3, 1, 2, 3, . . .
un

n

This divergent sequence is also periodic

Sequences and Series General Term of a Sequence Some sequences can also be defined by giving a general term. This general term is usually called the nth term. e.g. 1 e.g. 2 e.g. 3

2, 4, 6, 8, . . .

u n ? 2n

1 1 1 1 1, , , , . . . un ? 2 3 4 n

1, ? 4, 16, ? 64, . . . un ? ( ?4) n?1

The general term can easily be checked by substituting n = 1, n = 2, etc.

Sequences and Series Exercises 1. Write out the first 5 terms of the following sequences (a) (b) (c) (d)

un ? 1 ? 4n
un ? ( ?2) n u n ? 2n 2 un ? (?1) n

?3, ?2, 2, ?1,

? 7, ? 11, ? 15, ? 19 4, ? 8, 16, ? 32 8, 18, 32, 50 1, ? 1, 1, ? 1

2. Give the general term of each of the following sequences (a) 1, 3, 5, 7, . . . un ? 2n ? 1 (b) 1, 4, 9, 16, 25, . . . u ? n2 (c) ?3, 9, ? 27, 81, ? 243, . . .
n

un ? (?3) n

Series When the terms of a sequence are added, we get a series The sequence 1, 4, 9, 16, 25, . . .
gives the series 1 ? 4 ? 9 ? 16 ? 25 ? . . . Sigma Notation for a Series

Sequences and Series

A series can be described using the general term e.g. 1 ? 4 ? 9 ? 16 ? 25 ? . . . ? 100 can be written

?
1

10

n2

last value of n 1st value of n

? is the Greek capital letter S, used for Sum

Sequences and Series Exercises 1. Write out the first 3 terms and the last term of the series given below in sigma notation (a)

? 2n ? 1
1
100 1

20

? 1 ? 3 ? 5 ? . . . ? 39 n =n 1= 2 n = 20
? ? 3 ? 9 ? 27 ? . . . ? 3100

(b)

n ? ? ? ?3

2. Write the following using sigma notation (a) 2 ? 4 ? 6 ? 8 ? . . . (b) 2 ? 4 ? 8 ? . .

? 2n . ? 1024 ? ? ?2?
? 16 ?
10 1 n 1

8

Sequences and Series


推荐相关:


国际预科课程 (IFY).pdf

国际预科课程 (IFY) - NCUK 国际预科课程 (IFY) NCUK 国际本科预科课程(NCUK-IFY)是一个模块化的课程项目,旨在帮助出 国留学人员为大学本科学位课程做准备。...


Uploadify API详解.doc

Uploadify API详解 - JQuery 上传插件 Uploadify API 详解 一、相关 key 值介绍 uploader:uploadify.swf 文件的相对路径,该 swf ...


SAT词汇研究:后缀-ate 与-ify的具体介绍_教育指南_百度教育攻略.pdf

amplify v.放大,扩大(ampl大+ify→扩大) classify v.分类(class等级,类别+ify→分类) 可见,-ify很喜欢加在词根后面,表示更高级的意思。 ...


ThinkPHP中Uploadify文件上传返回HTTP Error(404)错误....doc

ThinkPHP 中 Uploadify 文件上传返回 HTTPError(404)错误解决办法: Uploadify 上传某些文档,如某些.docx 文档,返回的 type 值如下: application/vnd.openxml...


IFY国际基础课程简介.doc

IFY 国际基础课程简介 国际基础课程(IFY) , IFY 既 INTERNATION FOUNDATION YEAR,IFY 是由英国北方大学 联合会(NCUK)开发,专门为准备到英国大学联盟(UKUP)及澳洲...


IFY课程简介.doc

碧桂园教育集团 www.bgyedu.cn IFY 课程简介 IFY(Inter


JSON.stringify 语法实例讲解.doc

JSON.stringify 语法实例讲解_计算机软件及应用_IT/计算机_专业资料。JSON.stringify 语法实例讲解可能有些人对系列化这个词过敏,我的理解很简单。就是说把原来是...


关于uploadify的原理.doc

关于uploadify的原理 - Uploadify 在 Asp.net 中的使用 Uploadify 是 JQuery 的一个上传插件,实现的效果非常不错,带进度显示。首先按下面的 步骤来实现...


JQuery上传插件Uploadify3.1中文超级详细参考.doc

JQuery上传插件Uploadify3.1中文超级详细参考_计算机软件及应用_IT/计算机_专业资料。JQuery上传插件Uploadify3.1中文超级详细参考,JQuery,Uploadify,文件上传,异步上传,...


Uploadify-3.1-官方文档doc版.doc

Uploadify 3.1 官方文档 doc 版 Methonds Events


jquery上传插件Uploadify3.2中文详细参考手册.pdf

jquery上传插件Uploadify3.2中文详细参考手册 - 1. Uploadify 配置选项: ? auto 类型:Boolen 缺省值:true 说明:表示在选择文件后是否自动上传 ?...


uploadify教程.doc

uploadify教程 - Uploadify是一个jQuery插件,集成了一个


JQuery上传插件Uploadify使用详解 - 冯威的学习专栏---....pdf

//www.cnblogs.com/oec2003/archive/2010/01/06/1640027.html[2010-9-13 13:31:24] JQuery上传插件Uploadify使用详解 - 冯威的学习专栏---记录工作学习点滴 ...


SAT词汇之-ate、-ify的研究_教育指南_百度教育攻略.pdf

amplify v.放大,扩大(ampl大+ify→扩大) classify v.分类(class等级,类别+ify→分类) 可见,-ify很喜欢加在词根后面,表示更高级的意思。本...


Uploadify Version 3.2.1中文API文档.doc

Uploadify Version 3.2.1中文API文档_计算机软件及应用_IT/计算机_专业资料。Uploadify是一个免费的优秀的基于jQuery的插件,本文档使用官方提供的文档进行翻译并针对...


uploadify3.2+Struts2实现多文件上传显示进度条(及uplo....doc

uploadify3.2+Struts2实现多文件上传显示进度条(及uploadify详细介绍)_计算机软件及应用_IT/计算机_专业资料。uploadify3.2+struts2现实文件上传 ...


华南师范大学(IFY)国际留学预科.doc

华南师范大学(IFY)国际留学预科 - 华南师范大学国际本科留学预科班,是由华师


Uploadify插件用法.doc

Uploadify插件用法_计算机软件及应用_IT/计算机_专业资料。Uploadify插件用法,使用Uploadify须知 上传控件: 引用JS <script src='<% =ResolveUrl("~/JS/Uploadify...


NCUK IFY与其他预科课程及资格认证的不同之处.pdf

NCUK IFY 与其他预科课程及资格认证的不同之处 1.保证安置去向 IFY 项目保证:只要通过课程学习,即可被安置到 NCUK 的十一 所合作大学之中攻读学位。 由英国的...

网站首页 | 网站地图
All rights reserved Powered by 酷我资料网 koorio.com
copyright ©right 2014-2019。
文档资料库内容来自网络,如有侵犯请联系客服。zhit325@126.com