數(shù)列極限42 課件

1、2019/3/19,微積分,數(shù)列極限,1,上,課,2019/3/19,微積分,數(shù)列極限,2,第,2,章,極限與連續(xù),2019/3/19,微積分,數(shù)列極限,3,一、數(shù)列概念,數(shù)列,可看作自變量為正整數(shù)的,函數(shù),下標(biāo)函數(shù),n,y,u,f,n,2.1,數(shù)列的極限,2,特性,1,有界性,2,單調(diào)性,1,定義,按自然數(shù)編號依次排列的一列數(shù),1,2,n,u,u,u,稱為無窮數(shù)列,簡稱數(shù)列,記為,u,n,其中的每個(gè)數(shù),稱為數(shù)列的項(xiàng),u,n,稱為通項(xiàng),一般項(xiàng),數(shù)列,u,n,若,正數(shù),M,使得一切自然數(shù),n,恒,有,稱數(shù)列,u,n,有界,否則,稱為無界,n,u,M,1,2,n,u,u,u,稱此數(shù)列單調(diào)增加,1,

2、2,n,u,u,u,稱此數(shù)列單調(diào)減少,2019/3/19,微積分,數(shù)列極限,4,割之彌細(xì)，所,失彌少，割之又,割，以至于不可,割，則與圓周合,體而無所失矣,1,早期極限思想的體現(xiàn),放映,1,二、數(shù)列極限概念,當(dāng)自變量,n,趨于無窮大時(shí)，數(shù)列,y,f,n,的變化趨勢,y,n,1,劉徽的割圓術(shù),極限：研究函數(shù)在自變量的某個(gè)變化過程中，函,數(shù)值無限趨近于某個(gè)常數(shù)的性質(zhì),對于數(shù)列,2019/3/19,微積分,數(shù)列極限,5,R,正六邊形的面積,1,A,正十二邊形的面積,2,A,正,形的面積,1,2,6,n,n,A,3,2,1,n,A,A,A,A,S,2019/3/19,微積分,數(shù)列極限,6,2,莊子的截

3、丈問題,1,2,2,1,2,1,2,n,第一天剩余,u,1,第二天剩余,u,2,第,n,天剩余,u,n,1,2,n,0,但,0,n,時(shí),u,n,一尺之棰,日取其半,萬世不竭,1,1,1,1,2,4,8,1,6,1,0,1,8,1,2,1,n,2,1,0,2,1,n,n,時(shí),當(dāng),3,1,2,4,2,3,4,5,2,0,1,2,1,3,2,4,3,1,n,n,1,1,n,n,n,時(shí),當(dāng),3,1,1,1,1,0,1,1,間,擺,與,在,時(shí),當(dāng),1,1,1,1,n,n,1,4,l,i,m,n,n,n,ua,u,a,n,或,4,3,6,1,2,2,4,2,直觀定義,數(shù)列,u,n,若當(dāng),n,無限增大時(shí),u

4、,n,無限趨,近于常數(shù),a,則稱數(shù)列,u,n,以,a,為極限,或稱,u,n,收斂,于,a,記,1,2,n,n,u,1,n,n,n,u,1,1,n,n,u,1,3,2,n,n,u,1,3,2,n,n,發(fā),散,無限增大,例,否則稱,u,n,發(fā)散,2019/3/19,微積分,數(shù)列極限,8,1,1,1,時(shí)的變,當(dāng),觀察數(shù)列,n,n,n,播放,對于較簡單的數(shù)列的極限,可通過觀察法求得，例,1,1,5,1,1,l,i,m,l,i,m,2,l,i,m,l,n,1,l,i,m,l,i,m,1,n,n,n,n,n,n,n,n,n,e,n,n,n,n,n,0,2,0,1,1,lim,1,n,n,n,0,數(shù)列,極限

5、,的,嚴(yán)格,定義,2019/3/19,微積分,數(shù)列極限,9,1,1,1,1,n,n,n,u,n,當(dāng),無,限,增,大,時(shí),無,限,接,近,于,問題,無限接近”意味著什么,如何用數(shù)學(xué)語言刻劃,它,1,n,u,n,n,n,1,1,1,1,100,1,給定,100,1,1,n,由,100,時(shí),只要,n,1,1,1,0,0,n,u,有,1000,1,給定,1000,時(shí),只要,n,1,1,1,0,0,0,0,n,u,有,10000,1,給定,10000,時(shí),只要,n,1,1,1,0,0,0,n,u,有,0,給定,1,時(shí),只要,N,n,1,n,u,有,成,立,10,3.,N,定義,例,1,1,1,lim,1

6、,n,n,n,n,證明,證,1,1,1,n,n,n,n,1,0,1,n,u,要,1,n,只,要,1,N,取,1,1,1,n,n,n,有,1,1,l,i,m,1,n,n,n,n,故,l,i,m,0,d,n,n,n,ua,N,n,N,u,a,l,i,m,n,n,n,ua,u,a,n,或,設(shè)有數(shù)列,u,n,若對任意,總,則稱,a,是數(shù)列,u,n,的極限，或稱,u,n,收斂于,a,記作,0,存在正整數(shù),N,使得當(dāng),n,N,時(shí)，恒有,n,u,a,成立,否則稱數(shù)列,u,n,發(fā)散,則當(dāng),n,N,時(shí),11,注,3,N,一般與任意給定的正數(shù),有關(guān),越小,N,越大,例,2,l,i,m,n,n,n,u,C,C,u,

7、C,設(shè),為,常,數(shù),證,明,證,n,u,C,C,C,成立,0,任給,0,n,對于一切自然數(shù),l,i,m,n,n,u,C,故,說明,常數(shù)列的極限等于同一常數(shù),n,u,a,1,具有二重性,任意性和不變性。在取,時(shí),對其大小,不加限制，正由于這種,任意,性，才能用,刻劃,u,n,與,a,任意接近。而在根據(jù),找,N,時(shí)它是,不變,的,2,刻劃,u,n,與,a,接近的程度,N,刻劃數(shù)列作為動(dòng)點(diǎn)運(yùn)動(dòng)到什,么時(shí)刻可使,u,n,與,a,接近程度小于給定的,若把數(shù)列看成函,數(shù),則,N,分別用來刻劃因變量及自變量的變化過程,4,N,是不唯一的，用定義證明數(shù)列極限時(shí),關(guān)鍵是對任意,給定的,0,由,來,尋找,N,但不

8、必要求最小的,N,n,u,a,例,3,1,lim,0,2,n,n,證,明,證,1,0,2,n,1,2,n,0,0,n,u,要,2,1,log,n,只,要,2,1,log,N,取,1,0,2,n,有,1,lim,0,2,n,n,故,1,2,n,即,1,1,N,1,n,不妨設(shè),1,則當(dāng),n,N,時(shí),1,0,2,n,1,2,n,0,n,u,要,例,3,可用,放大手法,1,n,只,要,1,N,取,P31,例,2,取,要,設(shè),1,注,1,放大”是為方便解不等式。注意不能,放過頭,上,例,若將,放大為,1,則,1,不可能小于任意給定的正數(shù),1,2,n,2,放大”后找到的,N,通常比不放大解得,若易解,的要

9、大,2019/3/19,微積分,數(shù)列極限,13,x,1,u,2,u,1,N,u,3,u,2,a,a,a,2,N,u,三、數(shù)列極限的幾何意義,l,i,m,0,n,n,ua,N,則,正,整,數(shù),使得數(shù)列從,a,a,a,的,鄰,域,內(nèi),第,N,1,項(xiàng)起，以后所有項(xiàng),u,n,1,u,n,2,都落在,至多只有,N,項(xiàng)落在該,鄰域之外,l,i,m,0,d,n,n,n,ua,N,n,N,u,a,n,a,u,a,2019/3/19,微積分,數(shù)列極限,14,1,唯一性,定理,每個(gè)收斂的數(shù)列只有一個(gè)極限,證,l,i,m,l,i,m,n,n,n,n,u,a,u,b,設(shè),又,由定義,1,2,0,N,N,使,得,1,n

10、,nN,u,a,當(dāng),時(shí),恒,有,2,n,nN,u,b,當(dāng),時(shí),恒,有,max,2,1,N,N,N,取,時(shí)有,則當(dāng),N,n,n,n,ab,u,b,u,a,n,n,u,b,u,a,2,時(shí)才能成立,上式僅當(dāng),b,a,故收斂數(shù)列極限唯一,四、數(shù)列極限的性質(zhì),2019/3/19,微積分,數(shù)列極限,15,收斂數(shù)列的有界性的圖示,定理,收斂的數(shù)列必定有界,2,有界性,例如,1,n,n,u,n,數(shù),列,2,n,n,u,數(shù),列,有界,無界,數(shù)軸上有界數(shù)列的點(diǎn),u,n,都落在閉區(qū)間,M,M,上,2019/3/19,微積分,數(shù)列極限,16,證,l,i,m,n,n,u,a,設(shè),由定義,1,取,1,n,N,n,N,u,

11、a,則,使,得,當(dāng),時(shí),恒,有,1,1,n,a,u,a,即,有,1,m,a,x,1,1,N,M,u,u,a,a,記,n,n,u,M,則,對,一,切,自,然,數(shù),皆,有,n,u,故,有,界,注意,有界性是數(shù)列收斂的必要條件,推論,無界數(shù)列必定發(fā)散,定理,收斂的數(shù)列必定有界,l,i,m,n,n,u,a,設(shè),l,i,m,0,d,n,n,n,ua,N,n,N,u,a,n,a,u,a,取,1,則,n,N,時(shí),u,n,有界,2019/3/19,微積分,數(shù)列極限,17,五,小結(jié),數(shù)列,研究其變化規(guī)律,數(shù)列極限,極限思想,精確定義,幾何意義,收斂數(shù)列的性質(zhì),唯一性、有界性,思考題,1,試判斷下列論斷是否正確,1,若,n,越大,越接近于零,則有,n,u,a,lim,n,n,u,a,3,若對,存在自然數(shù),N,當(dāng),n,N,時(shí),數(shù)列,u,n,中,有無窮多項(xiàng)滿足不等式,則有,0,n,u,a,lim,n,n,u,a,2,若,則,n,越大,越接近于零,n,u,a,lim,n,n,u,a,4,若對,數(shù)列,u,n,中除了有限項(xiàng)外都滿足不,等式,則有,0,n,u,a,lim,n,n,u,a,3,從幾何直觀層次思考：若數(shù)列為單調(diào)