Re: [微積] 數列極限問題2則
※ 引述《wheniam64 (嘿)》之銘言：
: 1. Suppose that an →L. Show that if an 小於等於 M for all n,
: then L 小於等於 M .
For any ε ＞ 0, we have L–ε ＜ a ≦ M for some n, so L ≦ M.
n
(If a–ε ＜ b for any ε ＞ 0, then a ≦ b.)
: 2. Let f be a function continuous everywhere and let r be a real number.
: Define a sequence as follows:
: a1=r , a2=f(r) , a3=f(f(r)) , .........
: Prove that if an→L, then L is a fixed point of f:f(L)=L
Note that a ＝ f(a ).
n+1 n
L ＝ ｌｉｍ a ＝ ｌｉｍ f(a ) ＝ f(ｌｉｍ a ) ＝ f(L).
n→∞ n+1 n→∞ n ↖ n→∞ n
＼
f is continuous.
[Extension]
Moreover, if │f'(x)│＜ 1, for all x in [a,b] then
1. f has one and only one fixed point on [a,b].
2. {a } converges to the fixed point for any r in [a,b].
n
: 這二題習題和同學討論許久不得其解
: 請版上高手解答
: 感謝!
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【板主:dishpan/hayamavic/elisama】第十四屆小天使招生中 看板《NewAngels》
作者 NTUSTking (洪興帥哥 山雞) 站內 NewAngels
標題 [新生] NTUSTking
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08/17 23:06,
08/17 23:06
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※ 編輯: Minkowski 來自: 140.123.62.134 (08/17 23:51)
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08/17 23:55, 1F
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