Re: [分析] 請教一題複雜的連鎖題
: 思考了一段時間 覺得還是超出自己能力範圍
: 可能還是需要數學高手協助或者給予提示
: 希望有版上數學好手支援 謝謝
(本篇因為有大量運算排版，建議手機使用者橫向閱讀)
這裡只證明第一小題，後面幾小題看了一下應該都滿簡單的（？）
Let s_n = a_n + s, for n = 0, 1, 2, ... . -------(*)
The original statement is equivalent to proving
\lim_{n \to \infty} (a_0 + a_1 + ... + a_n)/(n+1) = 0
provided that \lim_{n \to \infty} a_n = 0.
Assume \lim_{n \to \infty} a_n = 0.
Since \lim_{n \to \infty} a_n = 0, given any \epsilon > 0,
there exists a positive integer N_1 s.t.
for all n > N_1 |a_n| < \epsilon/2. -------(A)
For such fixed N_1, we can find another positive integer N_2 > N_1 s.t.
|a_0 + a_1 + ... + a_{N_1}|/(n+1) < \epsilon/2. -------(B)
In other words, given any \epsilon > 0, for n > N_2
(a_0 + a_1 + ... + a_n)/(n+1)
= (a_0 + a_1 + ... + a_{N_1})/(n+1) + (a_{N_1+1} + a_{N_1+2} + ... + a_n)/(n+1)
<= |a_0 + a_1 + ... + a_{N_1}|/(n+1)
+ (|a_{N_1+1}| + |a_{N_1+2}| + ... + |a_n|)/(n+1)
< \epsilon/2 + (n-N_1)/(n+1) \epsilon/2
^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^
from condition (B) from condition (A)
< \epsilon/2 + \epsilon/2
= \epsilon.
We've established if \lim_{n \to \infty} a_n = 0,
then \lim_{n \to \infty} (a_0 + a_1 + ... + a_n)/(n+1) = 0,
giving our desired result that if \lim_{n \to \infty} s_n = s (by (*)),
\lim_{n \to \infty} (s_0 + s_1 + ... + s_n)/(n+1) = s.
有任何問題的話，歡迎有心的網友們一起來討論^^
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※ 編輯: cuylerLin (60.250.230.253 臺灣), 05/08/2020 20:42:40
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