【北京大學公開課】數(shù)學分析（上下全112講無級數(shù)部分）
第七講：無窮小序列
有界和無窮小序列：
引理：無窮小序列是有界性序列。
證明：先證明尾巴有界，再證明尾巴+前面也有界。
定理：
- 有界序列的和和乘積、差都有界。
- 無窮小序列之和是無窮小序列。
- 無窮小序列與有界序列的乘積也是無窮小序列
- 如果一個序列是無窮小序列，那么他的絕對值也是無窮小序列
- 兩個無窮小序列的乘積也是無窮小序列（把一個作為有界序列即可證明）
- 如果一個序列是無窮小序列，那么他乘以C之后，也是無窮小序列
有限個無窮小序列之和是無窮小序列
有限個無窮小序列之乘積也是無窮小序列
舉例：
b>1, any k in N, xn=n^k/b^n是無窮小序列。
隱含意義：任何指數(shù)增加都比冪次增加速度快。
下一個例子：冪次和階乘的對比
yn=c^n/n!, c>0.
=> yn infinitesimal sequence (inf. seq)
Exp:
alphan, inf. seq
betan=(alpha1+...+alphan)/n
=>betan inf.seq
Exp:
alpha inf. seq.
gamman=(alpha1*...*alphan)^(1/n)
gamman inf.seq.
Exp:
zn=(1/n!)^(1/n), zn inf.seq.
等價敘述法：
有界性的敘述：
- 存在M正，st any n, abs(xn) <=M. =>xn bounded.
- exists M, any n < M, abs(xn)<M => xn bounded.
無窮小序列等價敘述：
- 定義：any epsilon>0, exists N, s.t any n > N, abs(xn)<epsilon
- 等價：exists N', and n>=N', abs(xn)<epsilon
- abs(xn)<epsilon <=> abs(xn) <= epsilon
無窮下序列講到這里，下面開始講極限：
收斂序列（序列極限）
無窮小序列就是極限為0的序列。
(any epsilon>0, exists N, any n >N, abs(xn)<epsilon)
如果，存在a，any epsilon >0, exists N, any n>N, abs(xn-a)<epsilon.
=> xn converges to a. a is the limit of xn.
（epsilon <=> eps)
如果a不是xn的極限，那么，存在一個eps大于0，對所有N，都存在n>N，abs(xn-a)>eps
xn不收斂的定義：
對于所有的a, a is not limit of xn.
any a in R, exists eps>0, any N, exists n>N, abs(xn-a)>eps. (eps is related with a)
下面是一些定理：
if xn has limit, then the limit is unique.
a convergent (cvg) seq is bounded.
sandwich theorem:
xn<=yn<=zn for any n>= some N0, limxn=limzn=a,=>limyn=a.
四則運算：
lim xn=a, lim yn=b =>:
- lim(xn+-yn)=a+-b (=limxn+-limyn)
- lim(xn*yn)=a*b
- b!=0, lim(xn/yn)=a/b.
Corollary:
xn, real seq. the followings are equivalent.
- limxn=a
- xn-a is inf. seq
- ???
exp:
- lim(n/1+n)=1
- lim (n2-n+2)/3n2+2n+4=1/3
- a>1, w.t.s: (want to show) lima^(1/n)=1.
- 0<a<1. w.t.s lima^(1/n)=1.
- w.t.s limn^(1/n)=1.
Lecture 9: sequence limit
lim (n^k)^(1/n)=1.
exp:
- lim(n2+n)^0.5-n=1/2
if limxn=a, then lim(x1+...+xn)/n=a.
exp:
- lim(c+1/n)^(1/n), c>0.
- c<c+1/n<c+1
- lim sum_k=1^n q^(k-1), (abs(q)<1).
- lim an=A>0, an>0. w.t.s lim(a1*...*an)^(1/n)=A.
- lim xn=a, lim yn=b. cn=(x1yn+x2yn-1+...+xny1)/n. w.ts limcn=ab.
學過的四則運算都是對于有限位小數(shù)的，無限位小數(shù)的四則運算是怎樣的=>其實就是有限小數(shù)的極限的四則運算。
x=a0.a1a2...
y=b0.b1b2...
xn=a0.a1...an, yn=b0.b1...bn
xn'=x*10^n/10^n + 1/10^n
yn'=y*10^n/10^n + 1/10^n
xn<=x<=xn', same for y.
x+-y = lim(xn+-yn)
x*y=lim(xn*yn)
x/y=lim(xn/yn)
Properties of convergent sequences
cvg seq & inequalities
limxn=a, limyn=b, a<b. exists N, any n>N, xn<yn.
xn equiv a, large n, yn>a
Theorem: exists N0, any n>N0, xn<=yn. => limxn<limyn.
Remark: if xn<yn, !=> limxn<limyn.
xn<=zn<=y, any n>N0, limxn<=limzn<=limyn.
exp:
- a>=, b>0, w.t.s: lim(a^n*b^n)^(1/n)=max(a,b)
Lecture 10: Sequence comparison
exp:
- k>=2, large n, n^k, k^n, n!.
lim nk/kn=0, limkn/n!=0
- sign of (an2+ bn+c)/An2+Bn+C)
convergence theorem
limxn=a, when a is unknown, how to show xn is cvg?
Theorem: monotonic seq:
mono increasing, upper bounded, xn has a limit.
mono decreasing, lower bounded, xn has a limit.
