Let {U(n)} be a sequence of real numbers such that:
U(1) = k, where k is real and is not equal to √2, -1 or -√2
U(n+1)=[U(n)+2]/[U(n)+1] for n=1, 2, 3, ...
It is known that if the sequence converges, and has a limit of L,
L=(L+2)/(L+1)
L^2+L=L+2
L=√2 or -√2
Assume that U(n) is not equal -1 for n = 1, 2, 3, ...,
explain whether there exists some values for k such that
(i) lim (n--> infinity) U(n) = √2
(ii) lim (n--> infinity) U(n) = -√2
If possible, find the general term U(n) of this sequence in terms of k.
Limit of sequence
2010-07-16 5:18 am
回答 (2)
2010-07-23 6:59 am
✔ 最佳答案
最後一題未代U(1) = k時的答案（由於wolfram alpha判斷的限制，故只能把U換成a）：http://www.wolframalpha.com/input/?i=a%28n%2B1%29a%28n%29%2Ba%28n%2B1%29%3Da%28n%29%2B2
至於過程則未明，因為我完全看不明白http://eqworld.ipmnet.ru/en/solutions/fe/fe2301.pdf和http://eqworld.ipmnet.ru/en/solutions/fe/fe2314.pdf。
2010-07-15 22:37:23 補充：
我相信最後一題的解法與http://tw.knowledge.yahoo.com/question/question?qid=1610062604007類似。
2010-07-21 15:31:55 補充：
這些題目有沒有分先後次序做？
即是可否先find the general term U(n)然後才做(i)(ii)部，還是一定要先做(i)(ii)部才find the general term U(n)？
2010-07-21 22:25:46 補充：
點解就連解題的先後次序都可以自己自由決定？莫非這題不是功課題而是你自己出的題目？
2010-07-21 22:36:36 補充：
U(n+1)=[U(n)+2]/[U(n)+1]
U(n+1)=1+1/[U(n)+1]
Let a(n)=U(n)+1,
Then a(n+1)=U(n+1)+1
∴a(n+1)-1=1+1/a(n)
a(n+1)=2+1/a(n)
正是http://tw.knowledge.yahoo.com/question/question?qid=1610062604007的類型
2010-07-22 22:59:04 補充：
圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/differenceequation/diffequanalysis9.jpg
參考資料：
my wisdom of maths
2010-07-16 7:07 am
For
http://tw.knowledge.yahoo.com/question/question?qid=1610062604007
we just need to define:
f(n)=a(1)a(2)...a(n)
Then,
a(n)=α+β - (αβ)/a(n-1)
f(n)=(α+β)f(n-1) - (αβ)f(n-2)
which is a commonly seen difference eqt.
Thx for the links, doraemonpaul. They are useful.
2010-07-21 22:04:14 補充：
I think it is better to do (i)(ii) first, i.e. whether the fixed pts √2 and - √2 are stable or unstable.
You can find the general term first if you want to.
2010-07-21 23:14:01 補充：
I have tried several k s, but it seems that U(n) never tends to - √2. I just guess - √2 cannot be a limit and ask this question to seek some mathmatical proof for proving the limit can never be - √2.
http://tw.knowledge.yahoo.com/question/question?qid=1610062604007
we just need to define:
f(n)=a(1)a(2)...a(n)
Then,
a(n)=α+β - (αβ)/a(n-1)
f(n)=(α+β)f(n-1) - (αβ)f(n-2)
which is a commonly seen difference eqt.
Thx for the links, doraemonpaul. They are useful.
2010-07-21 22:04:14 補充：
I think it is better to do (i)(ii) first, i.e. whether the fixed pts √2 and - √2 are stable or unstable.
You can find the general term first if you want to.
2010-07-21 23:14:01 補充：
I have tried several k s, but it seems that U(n) never tends to - √2. I just guess - √2 cannot be a limit and ask this question to seek some mathmatical proof for proving the limit can never be - √2.
收錄日期: 2021-04-30 01:16:33
原文連結 [永久失效]:
http://hk.knowledge.yahoo.com/question/question?qid=7010071501617