5BM1U3-2

u1 (u2 u3 ) (u2m2 u2m1 ) (u2m u2m1 )
S2m1 即数列 {S2m-1 } 单调减少, 又因
un1 un 0,
中央财经大学
数学分析
S2m1 (u1 u2 ) (u3 u4 ) (u2m3 u2m2 ) u2m1
(u1 u2 )
[sin(n 1)x sin(n 1 )x]
2
2
sin(n
1 )x
2
当
x (0,2 )
时,
x sin
0,
故得到
2
1
1
n
sin(n x)
cos kx
2
2 k1
2sin x
2
中央财经大学
数学分析
所以级数 cosnx 的部分和数列当 x (0,2 ) 时 有界,由狄利克雷判别法推得级数 an cosnx 收敛. 同理可证级数 an sinnx 也是收敛的.
证明：由阿贝尔变换
同号
m
m1
S aibi | (ai ai1) || Bi | | amBm |
i1
i1
m1
S M | (ai ai1) | | am | M i 1
m
故 S aibi M ( a1 2 am ) i1
中央财经大学
数学分析
三、阿贝尔判别法和狄利克雷判别法
阿贝尔判别法
数学分析
第九章 级数
数项级数
中央财经大学
数学分析
III 任意项级数
一、交错级数及其审敛法
定义: 正、负项相间的级数称为交错级数.
(1)n1un或 (1)nun (其中un 0)
n1
n1
莱布尼茨定理 如果交错级数满足条件:

项级数, 且存在某正数 N0 及常数l,
(i) 若对一切 n N0, 成立不等式
n un l 1,
(9)
则级数 un 收敛;
(ii) 若对一切 n N0, 成立不等式
n un 1,
(10)
则级数 un 发散.
33
证 由(9)式有un l n , 因为等比级数 l n 当 1 l 1 时收敛, 故由比较原则, 这时级数 un 也收敛, 对
发散.
(3)
lim un1 lim (2n 1) 2n
1,
u n n
n (2n 1) (2n 2)
比值审敛法失效, 改用比较审敛法
(2n
1 1) 2n
1 n2
,
级数
n 1
1 n2
收敛,
故级数
1 收敛.
n1 2n (2n 1)
26
例6 级数
2 2 5 2 5 8 2 5 8 [2 3(n 1)] ,
11 2
1 2 2n1
2,
sn 有上界,
1 .
n1 n!
5
Ex2 : 正项级数
n1
1 np
1
1 2p
1 3p
1 np
称为p 级数(广义调和级数),讨论其敛散性.
解 :1 . 当p 1时, p 级数正好是调和级数
1 ,其部分和 n 1 1+ 1 1 1 无上界.
n1 n
定理).这就证明了定理的结论.
4
注: (1)叙述基本定理的逆否命题.
(2)正项级数敛散性的所有的判别法, 归根到底,都是根据这条简单的定理.
Ex1: 证明: 1 n1 n!
证:
1 1 n! 12

各位老师，提供的教材分析比较全面，但仅供大家参考，在使用过程中一定要学会唯教师所有、唯学生所用。

此外，不要忘记参照教学参考资料提供的所有内容，包括提供的单课时活动设计、教学小贴士等等。

浦东教发院吴建新Oxford English Book 5B Module 1Using my five senses教材分析提供：邱烨红张慧慈丁征宇庄丽华（张江高科实验小学）审核统稿：吴建新（浦东教发院）一、模块教学任务分析主题：Using my five senses单元：Unit 1 What a mess! Unit 2 Watch it grow! Unit 3 How noisy!5BM1的学习主题为Using my five senses。

这个模块涉及到的新内容包括学校里经常使用的一些学习物品（如笔记本、蜡笔、胶带、颜料等），名词性物主代词（如mine, yours, his, hers, ours, theirs等），昆虫、动物成长过程的不同阶段名称（如egg, cocoon, caterpillar, puppy等），能发出特殊响声的一些生活物品类单词（如drill, lorry, motorbike等）。

在学习本模块之前已经涉及了不少人称代词的主格、宾格和所有格（又称形容词性物主代词）、一些常见的学习物品、某些常见的昆虫和动物的名称以及一些常见的交通工具等等。

这些都是学习本模块内容的基础和前提。

本模块的句型与时态，主要是进一步复习和巩固一般现在时态和can句型。

在此基础上，出现一种全新的时态，即一般过去时态(simple past tense)。

一般过去时态则从be动词的用法开始学习，转向行为动词的一般过去时态的陈述形式的用法与表达。

从身体感官为出发点，以“真是一团糟”、“昆虫、动物的成长”和“生活中的嘈杂声音”为话题展开学习内容，不仅注意到了新旧知识的有机整合，而且对学生的语言能力提出了更高的要求，进一步从听说读能力的培养转向听、说、读、写、练、用的全面兼顾。

上海4B牛津英语知识整理总结4B M1 U1一、词组，句子：1.a glass of watermelon juice 一杯西瓜汁2.taste some cherries 尝尝一些樱桃3.two glasses of grape juice 两杯葡萄汁4.look and guess 看一看猜一猜5.smell these strawberries 闻闻这些草莓6.much fruit 许多水果7.how about ...怎么样8.these crunchy plums 这些爽脆的李子9.those purple grapes 那些紫色的葡萄10.what nice grapes 多好的葡萄啊11.on the tall vine 在高高的葡萄藤上12.not ...at all 一点也不13.sit around 团团坐14.enjoy the fruit 享用水果15.in his yard 在他的的院子里16.his sister Sue 他的妹妹Sue17.on sunny afternoons 在阳光灿烂的下午18.go to school 上学19.a juicy watermelon for me 一个多汁的西瓜给我20.a crunchy pear for them 一个爽脆的梨给他们21.I think. 我认为。

22.How does the red apple taste? = How is the red apple? It's sweet.23.How do the red apples smell? = How are the red apples? They're nice.24.How about the green one? It's rough.25.How about the green ones? They're smooth.26.What nice grapes they are! = How nice the grapes are!27.The grapes are not sweet at all.28.Do you like the hard ones or the soft ones? I like the hard ones.29.He also rides the bicycle around the park on sunny afternoons.30.He never rides his bicycle when he goes to school.二、音标：1.ar---/ ɑ: / car card park Mark are hard yard bar farmfarmer market supermarket star party scarf sharpMarch far arm large garden smart cartoona----/ ɑ: / class glass basketball grass mask branch after afternoon past plant dance fast father France2. u ---- / u: / blue glue true Sue ruler ruleui ----/ u: / fruit juice juicyo----- / u: / do to who whoseoo----/ u: / afternoon school cool moon room smooth food tooth zoo soon spoon noodles cartoon choose tooroot balloon三、短文诵读：Sour grapesOne day, a fox is thirsty. He is looking for （寻找）some water. Then he sees some bunches of grapes on the tall vine. The grapes are purple and r ound. They’re smooth and soft. What nice grapes! The fox looks at the grapes. He thinks the grapes are sweet and juicy. He wants them. The fox jumps and jumps. But he can’t touch the grapes.Suddenly（突然）a bird is coming. She sees the grapes too. She flies to the grapes and eats the grapes happily. The fox is angry:“These are my grapes! Go away!”The bird says:“Haha! You can’t fly high and enjoy（享受）the nice grapes. So they are my grapes.”The fox is envious /'envi?s/ (嫉妒的). “Oh! Those grapes are sour. They are not sweet at all!” Then the fox goes away.4B M1 U2一、词组，句子：1.whose knife 谁的小刀2.these blunt knives 这些钝的小刀3.those sharp pencils 那些锋利的铅笔4.Jill's pencil case Jill的铅笔盒5.many thick books 许多厚的书6.some thin books 一些薄的书7.walk on the beach 在海边走8.Kitty and her parents Kitty和她的父母亲9.much soft sand 许多软的沙子10.take off her shoes 脱下她的鞋子11.put on their coats 穿上他们的外套12.of course 当然可以13.the key to the door 开门的钥匙14.the key to the question 问题的答案15.the man over there 在那里的男子16.take the key to the lost-property office 失物招领处17.the blind brothers 盲兄弟18.touch the elephant 碰碰象19.one of the pupils 学生中的一个20.another brother 另一个兄弟21.smooth and hard 又光滑又硬22.big and thick 又大又厚23.the last blind boy 最后一个失明的男孩24.say about the elephant 说说大象25.some fruit for him 给他的一些水果26.some drinks for her 给她的一些饮料27.in the school yard 在学校的院子里28.a slim girl 一个苗条的女孩29.a red new skirt 一条新的红裙子30.too small 太小31.find her purse 找到她的小包32.on the floor 在地板上33.behind the door 在门后34.want to do sth. 想要做某事35.her cousin Laura 她的表妹Laura36.give sth. to sb. = give sb. sth. 把某物给某人37.take sth. to sb. = take sb. sth. 拿某物给某人38.Give the thin book to Grandma. = Give Grandma the thin book.39.Take the key to the lost-property office.40.How does the pineapple feel? = How is the pineapple? It's rough and hard.41.How do the pillows feel? = How are the pillows? They're soft.42.Whose books are those? = Whose are those books? They're Miss Fang's.43.What's the matter with you? = What's wrong with you?44.There's something in the sand.45.Here comes a man on an elephant.46.The brothers can hear the elephant, but they cannot see it.47.' It's an elephant,' answers the man, ' You can touch it. How does it feel? '48.' It's over there,' says her mum, ' Just behind the door!'49.She wants to buy a new purse.50.He likes eating and drinking. = He likes to eat and drink.二、音标：1. ir /?:/shirt bird third thirsty first girl skirt birthdayur /?:/purse nurse Thursdayer /?:/her certainly(当然) term （学期）or /?:/work worker world wordear /?:/ early learn2. au /?:/ warm war quarter 一刻钟al /?:/ all ball call fall hall small tall walk always aw /?:/draw strawberry prawn虾au /?:/August autumn daughter 女儿because Laura oor /?:/door floorour /?:/your four fourteenore /?:/ more before4BM1 U3一、词组，句子：1.behind the hill 在小山丘后面2.the tree’s shadow 树的影子3.on the lawn 在草坪上4.at noon 正午5.high in the sky 高高挂在天上6.grow short 变短7.these benches 这些长凳8.go up = rise 升起9.go down = set 落下10.grow long again 又变长11.on the path 在小路上12.cut them out 把它们剪下来13.stick a pencil 黏上铅笔14.shine the torch 打开手电筒15.turn on the torches 打开手电筒16.at seven o’clock in the morning 在上午七点17.go with me 跟我走18.stay behind me 呆在我后面19.walk in front of him 走在我前面20.take a walk 散步21.go to work 上班22.grow big and strong 变得又大又强壮23. a beautiful sunny day 一个美丽的好天24.walk down the road 沿着小路向下走25.hear a sound 听见一种声音26.look back 回头看27. a black shape 一个黑色的图形28.bite her 咬她29.be afraid of the dog 害怕狗30.run away 逃走31.follow them 跟着他们32.play together 一起玩33.play with the shadow 玩影子34.run after 跟着跑，追赶35. a big black dog 一个大黑狗36.go off 消失37.not …any more 再也不38.have tea with his friend Miss Deer 与他的朋友Deer小姐一起喝茶39.fall into the pond 掉到池塘里40.cry in fear 害怕得哭41. a bottle of beer 一瓶啤酒42.burst into tears 突然哭起来43.burst into laughter 突然笑起来44.be like = look like 像…45.You won’t see it any more. 你再也看不见它。