Shephard 2017-12-9 18:21:05

:^(

點解dom(x^0) 會包埋0 ？

你邊度睇返嚟㗎

:^(

WolframAlpha, Symbolab

http://www.wolframalpha.com/input/?i=dom%28x%5E0%29

我覺得係WolframAlpha 錯

:^(

同感

:^(

Elias 2017-12-10 14:39:22 pish

蘋果薄荷 2017-12-10 20:21:15 留名學嘢

Edelschwarz 2017-12-14 03:03:06 食埋飯出文

其實仲有冇人睇緊

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:^(

:^(

畀個正皮點人頭得唔得

:^(

Petrushka 2017-12-14 03:09:04 好想做mature student入番去讀數
（啲中學師弟以為我當年走左去讀數，點鳩知係完全唔關事既科

:^(

）
LM學野

... 2017-12-14 03:11:23

:^(

Elias 2017-12-14 06:13:03

好想做mature student入番去讀數
（啲中學師弟以為我當年走左去讀數，點鳩知係完全唔關事既科

:^(

）
LM學野

做咩咁睇唔開

:^(

Penthrite 2017-12-14 07:18:44 有冇textbook 適合自學analysis? 最好由real 到functional 都有
有足夠algebra 底但係冇讀過任何analysis course

Edelschwarz 2017-12-14 08:05:54 講完sequence，我地終於有工具去討論一啲比較有趣嘅問題嚕

:^(

今次我地就講吓加無限多個數嘅問題
例如點解我地會話 1+1/2+1/4+1/8+... = 2

:^(

因為我地成日都會討論呢啲問題，所以我地又會畀個名佢真係好鬼多名要記

對於一條sequence {an} , 我地可以整一條新嘅sequence {sn}, sn嘅值就係頭n個term加埋嘅和，即係咁

咁呢條sequence就叫做sequence of partial sums

如果覺得太抽象，可以諗吓下面嘅例子:
對於 1/2, 1/4, 1/8, ... 呢條sequence,
佢對應嘅{sn}就會係 1/2, 3/4, 7/8, ...

咁呢條新嘅sequence，我地叫佢做一個級數(series)，通常就會用呢個符號代表

其實呢舊野係abuse of notation(濫用符號?)，不過是但啦

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如果呢條{sn} converge去L，咁我地就會話呢個series converge去L, 而呢個L就叫做sum of the series (級數和?

:^(

)，咁我地就通常咁寫:

今次冇咁abuse of notation喇

反之，如果{sn} diverge (to +/- infinity)，咁我地就話呢個series diverge (to +/- infinity)

有例子未

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我地首先要討論嘅就係geometric series (DSE嘅同學仔應該見過咩係geometric progression，呢個就係佢嘅無限版)
佢係咁樣樣嘅:

為咗方便，啲人通常由n=0開始數，但其實冇咩所謂

咁當r≠1嗰陣，我地嘅sn就會係咁嘅樣:


如果-1<r<1嘅話，咁rᵏ⁺¹就會越嚟越細，慢慢趨向0
所以個series就會converge去1/(1-r)

我地望返post頂嗰條series 1+1/2+1/4+1/8+...
咁佢其實係一條geometric series (c=1, r=1/2)
所以個series個sum就係1/(1-1/2) = 2

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如果|r|≥1, 咁點算呀

:^(

咁大家見到後面嘅term唔會變細，咁佢地加埋就梗係唔會converge啦

所以佢地就係divergent喇

我地由上面嘅例子見到，要一個series converge，咁{an}就一定要趨向0
睇落好啱，但係點prove呢?


如果一個series converge, 咁{sn}就係一條convegent sequence (by definition)，所以佢係一條Cauchy sequence
咁對於求其一個ϵ，我地都搵到一個N，使到第N個term之後任意兩個term嘅差都細過ϵ
咁當n>N嗰陣，代入返呢個結果，我地就得到
|sn-sn-1| = an<ϵ
所以{an} converge去0
搞掂

:^(

咁調返轉得唔得呢?
如果一條sequence係converge去0, 咁條series係咪一定converge呢?

唔係

Counterexample在此:

呢個可能係最出名嘅divergent series，叫做Harmonic series

點解佢diverge? 我地睇吓佢嘅partial sum:

如此類推，我地得到:

由此可見sn係冇上限嘅，所以佢diverge

:^(

知道佢diverge有咩用?

識少少physics嘅朋友仔可以計到如果我地有n舊長度為1嘅磚擺喺枱面，咁最頂嗰舊磚就可以伸到離張枱

咁遠

咁因為Harmonic series係divergent，所以我地只要有好多好多(真係好多)舊磚，最頂舊磚伸到幾遠都得

:^(

如果你想伸到100咁遠，你需要超過10^43舊磚

Shephard 2017-12-14 08:06:59

有冇textbook 適合自學analysis? 最好由real 到functional 都有
有足夠algebra 底但係冇讀過任何analysis course

https://g.co/kgs/nvvrJL

Elias 2017-12-14 08:32:16 divergence test就咁
lim x_{n +1} = lim (s_{n + 1} - s_n) = = L - L = 0
咪得

:^(

Edelschwarz 2017-12-14 09:40:09 跟住我地就講吓一啲分辨series係咪converge嘅方法 (convergence tests)

首先我地有Comparison test:

個idea其實好簡單
如果大嗰個series冇爆煲，咁細嗰個series都唔會爆煲
如果細嗰個series都爆煲，咁大嗰個series都會爆煲
Proof: Exercise

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提示: 可以對partial sum {sn}使用Cauchy sequence
做乜好似搞到打Pokémon咁

佢嘅好處係唔使計
壞處係你首先要知道邊啲series係converge先有得比較

一個常見嘅example就係p-series:


我地依家就試吓證明當p=2嘅時候，呢個series係convergent

首先我地要搵條可以比較嘅series:

點解咁樣衰嘅
咁我地發現，當我地將頭n個term加埋嘅時候，奇蹟發生了:

咁呢個series嘅partial sum就會converge去1

然後我地將個2-series重寫吓


我地就得到


咁comparison test就話畀我地知 2-series係convergent嘅
搞掂

:^(

其實comparison test仲話畀我地知

:^(

(諗吓點解?)


Exercise: 用comparison test去證明當p>2嘅時候，p-series係convergent


比較可惜嘅係呢個方法(同埋其他convergence tests)只可以話畀你知佢係咪converge，但係無辦法知道佢地converge去邊
例如我地知道

:^(

但係如果你搵得到3-series嘅sum嘅exact value，下屆Fields medal應該會係你攞

:^(

言歸正傳，我地講吓其他convergence test:


Proof: 如果r>1, 咁啲term就唔會converge去0, 所以個series就會diverge
如果r<1, 大家可以試吓同geometric series比較吓
即係exercise囉

:^(

例子: 讀過Taylor series嘅同學仔都應該知道呢個series

係convergent

而ratio test就話畀我地知


要注意嘅係如果r=1嘅話，呢個test係廢嘅(i.e. 無法提供任何資訊)
例如對於Harmonic series同2-series，佢地個r都係1, 但係一個diverge一個converge

除咗Ratio test, 我地仲有個差唔多嘅test


Proof: 同ratio test個proof一樣

:^(

最後我地有一個好奇怪嘅test:


Proof:
我地考慮佢嘅partial sum {sn}
因為an係+ve同decreasing，我地可以推斷sn+1喺sn同sn-1之間(停一停，諗一諗)
於是我地知道(by induction) 當m>n嗰陣 sm喺sn同sn-1之間
而因為{an}趨向零，對於任何ϵ，我地都搵到個N令到|sN+1-sN|<ϵ
所以當m,n>N嗰陣，我地得到|sm-sn|<ϵ
所以{sn}係Cauchy sequence，所以converge
搞掂

:^(

例子:
呢個series係convergent:

各位記性好嘅話應該認得出呢個係我話明顯Cauchy嘅series (雖然望返好似又真係唔係好明顯)
希望大家對呢度上面嘅解釋滿意啦

:^(

下集: conditional convergence

忒修斯之船 2017-12-14 09:41:40

:^(

Edelschwarz 2017-12-14 09:42:36

divergence test就咁
lim x_{n +1} = lim (s_{n + 1} - s_n) = = L - L = 0
咪得

:^(

呢個方法如果個思維跳唔到
好似好難解釋到人地明
其實係我係跟我睇嘅notes，唔關我事㗎

Edelschwarz 2017-12-14 09:45:52

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

忒修斯之船 2017-12-14 09:47:55

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

Edelschwarz 2017-12-14 09:49:58

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

(under some conditions)

:^(

Elias 2017-12-14 09:50:43

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

:^(

Edelschwarz 2017-12-14 09:58:30

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

:^(

:^(

Elias 2017-12-14 10:00:37

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

:^(

:^(

:^(

Elias 2017-12-14 10:04:43

:^(

唔係有講我地用得越多term就會越準咩
咁有無限個term咪無限準(i.e. converge to true value)

:^(

:^(

:^(

:^(

:^(

:^(

:^(

:^(

會考都無C 2017-12-14 10:40:03 媽 好深呀

:^(

Elias 2017-12-14 10:55:57

媽 好深呀

:^(

analysis好恐怖架, 唔好讀math major呀

:^(

會考都無C 2017-12-14 11:01:22

媽 好深呀

:^(

analysis好恐怖架, 唔好讀math major呀

:^(

但係唔想鳩背project present essay lab嘅話仲有咩科可以揀

:^(

Elias 2017-12-14 11:04:29

媽 好深呀

:^(

analysis好恐怖架, 唔好讀math major呀

:^(

但係唔想鳩背project present essay lab嘅話仲有咩科可以揀

:^(

stat?