【北京大学公开课】数学分析（上下全112讲无级数部分）

第七讲：无穷小序列

有界和无穷小序列：

引理：无穷小序列是有界性序列。

证明：先证明尾巴有界，再证明尾巴+前面也有界。

定理：

1. 有界序列的和和乘积、差都有界。
2. 无穷小序列之和是无穷小序列。
3. 无穷小序列与有界序列的乘积也是无穷小序列
4. 如果一个序列是无穷小序列，那么他的绝对值也是无穷小序列
5. 两个无穷小序列的乘积也是无穷小序列（把一个作为有界序列即可证明）
6. 如果一个序列是无穷小序列，那么他乘以C之后，也是无穷小序列

有限个无穷小序列之和是无穷小序列

有限个无穷小序列之乘积也是无穷小序列

举例：

b>1, any k in N, xn=n^k/b^n是无穷小序列。

隐含意义：任何指数增加都比幂次增加速度快。

下一个例子：幂次和阶乘的对比

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.

等价叙述法：

有界性的叙述：

1. 存在M正，st any n, abs(xn) <=M. =>xn bounded.
2. exists M, any n < M, abs(xn)<M => xn bounded.

无穷小序列等价叙述：

1. 定义：any epsilon>0, exists N, s.t any n > N, abs(xn)<epsilon
2. 等价：exists N', and n>=N', abs(xn)<epsilon
3. abs(xn)<epsilon <=> abs(xn) <= epsilon

无穷下序列讲到这里，下面开始讲极限：

收敛序列（序列极限）

无穷小序列就是极限为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的极限，那么，存在一个eps大于0，对所有N，都存在n>N，abs(xn-a)>eps

xn不收敛的定义：

对于所有的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.

四则运算：

lim xn=a, lim yn=b =>:

1. lim(xn+-yn)=a+-b (=limxn+-limyn)
2. lim(xn*yn)=a*b
3. b!=0, lim(xn/yn)=a/b.

Corollary:

xn, real seq. the followings are equivalent.

1. limxn=a
2. xn-a is inf. seq
3. ???

exp:

1. lim(n/1+n)=1
2. lim (n2-n+2)/3n2+2n+4=1/3
3. a>1, w.t.s: (want to show) lima^(1/n)=1.
4. 0<a<1. w.t.s lima^(1/n)=1.
5. w.t.s limn^(1/n)=1.

Lecture 9: sequence limit

lim (n^k)^(1/n)=1.

exp:

1. lim(n2+n)^0.5-n=1/2

if limxn=a, then lim(x1+...+xn)/n=a.

exp:

1. lim(c+1/n)^(1/n), c>0.
2. c<c+1/n<c+1
3. lim sum_k=1^n q^(k-1), (abs(q)<1).
4. lim an=A>0, an>0. w.t.s lim(a1*...*an)^(1/n)=A.
5. lim xn=a, lim yn=b. cn=(x1yn+x2yn-1+...+xny1)/n. w.ts limcn=ab.

学过的四则运算都是对于有限位小数的，无限位小数的四则运算是怎样的=>其实就是有限小数的极限的四则运算。

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:

1. a>=, b>0, w.t.s: lim(a^n*b^n)^(1/n)=max(a,b)

Lecture 10: Sequence comparison

exp:

1. k>=2, large n, n^k, k^n, n!.

lim nk/kn=0, limkn/n!=0

1. 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.