Re: [微積] 高等微積分求高手幫忙
※ 引述《qoo11808 (qoo11808)》之銘言：
: 小弟已解出第一個，但另外的ㄧ個請各位高手幫忙。
: 要用到｛an} cov. to 0
: 老師好像說要跟1/k,epsilon比較。不太懂。
提供一種想法
For all ε> 0, there exist M > 0 s.t M > 1/ε.
Since 1/a_n --> ∞, there exists a positive integer N s.t
n > N => 1/a_n > M > 1/ε
=> a_n < ε
Hence a_n --> 0.
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.114.234.61
※ 文章網址: http://www.ptt.cc/bbs/Math/M.1411473605.A.A30.html
※ 編輯: Ryoui (140.114.234.61), 09/23/2014 20:05:02
推
09/23 20:06, , 1F
09/23 20:06, 1F
→
09/23 20:10, , 2F
09/23 20:10, 2F
→
09/23 20:10, , 3F
09/23 20:10, 3F
推
09/23 20:12, , 4F
09/23 20:12, 4F
→
09/23 20:13, , 5F
09/23 20:13, 5F
推
09/23 20:24, , 6F
09/23 20:24, 6F
推
09/23 20:27, , 7F
09/23 20:27, 7F
我猜他是要你們多做一步
把 1/a_n 用 b_n 表示
然後把 1/a_n --> ∞ 用 b_n --> ∞ 去套入定義去證明 a_n --> 0
不過我覺得很多餘就是了
※ 編輯: Ryoui (140.114.234.61), 09/23/2014 20:32:42
推
09/23 20:36, , 8F
09/23 20:36, 8F
→
09/23 20:36, , 9F
09/23 20:36, 9F
推
09/23 21:55, , 10F
09/23 21:55, 10F
→
09/23 21:59, , 11F
09/23 21:59, 11F
→
09/23 22:03, , 12F
09/23 22:03, 12F
推
09/23 22:08, , 13F
09/23 22:08, 13F
推
09/23 22:32, , 14F
09/23 22:32, 14F
我用你的語法寫一次 你在比較一下跟上面有什麼不同吧
Let b_n := 1/a_n --> ∞. By definition, we have
for all k > 0, there exists positive integer n*
(＊)
s.t b_n > k , for all n > n*
Now we show that a_n --> 0.
for all ε > 0, there exists k > 0
￣￣￣￣￣￣￣ (阿基米德公理)
s.t k > 1/ε
For this k, there exists a positive integer n* (從＊得到)
￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣￣
s.t b_n = 1/a_n > k > 1/ε, for all n > n*
￣￣
=> a_n < ε, for all n > n*
￣￣￣￣￣￣￣￣￣￣￣￣
Hence,
for all ε > 0, there exists a positive integer n*
s.t a_n < ε, for all n > n*
By definition, a_n --> 0.
※ 編輯: Ryoui (140.113.22.70), 09/24/2014 17:11:40
推
09/24 22:23, , 15F
09/24 22:23, 15F
討論串 (同標題文章)