四道数学分析的问题诚心求解！

abcboring123 · 发表于 2012-12-9 21:07

本人在国外上学，这是老师留的四道带回家期末的题目。诚心求教！如方便请把步骤写得相对详细和规范，本人基础相对较差。
如有语言上的问题请及时告知.
题目在附件的文档里。

elimqiu · 发表于 2012-12-10 12:36

[这个贴子最后由elimqiu在 2012/12/10 05:02pm 第 1 次编辑]

elimqiu · 发表于 2012-12-10 12:39

I guess you have to learn how to do ex.2. Otherwise you can hardly digest e-N language.

数学的 · 发表于 2012-12-10 12:39

第二题Stolz公式。其他好像利用定义

elimqiu · 发表于 2012-12-10 12:42

Ex.3
By mean value theorem, f(x+1) - f(x) = f';(w) for some w between x+1 and x.
Let x goes to infinite, w must go with it and so lim (f(x+1)-f(x)) = 0 as (x goes to infinite)

elimqiu · 发表于 2012-12-10 12:49

[这个贴子最后由elimqiu在 2012/12/10 05:38pm 第 1 次编辑]

Ex4 Hint
Given e > 0, there is a partition of [a,b], such that  U(P,f) - L(P, f) < e/2
Add some points to P that makes the sum S of the lengh of those [x_i, x_{i+1}] containing the finite points where f and g differ small enough such that S(M-m) < e/2 where M is the upper bound of g,
m is the lower bound of g and then show that
U(P*, g) - L(P*, g) < e where P* is the refined partition of P.

abcboring123 · 发表于 2012-12-10 22:27

第二题做出来了，非常感谢！其他题目我感觉楼上写的我还看得不是很懂。。。惭愧。。。您要是不介意能写的再详细和标准点吗？。。：（[br][br]-=-=-=-=- 以下内容由 abcboring123 在时添加 -=-=-=-=-
第三题也经过您的指点做出来了！！非常感谢！

elimqiu · 发表于 2012-12-11 00:13

I fixed a typo in Sol of Ex1.
pls tell me which line you don';t understand or find mistake.
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Ex.4. What is your Def. of integrable?

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