Re: 數列收斂三個問題
※ 引述《YmemY (**米)》之銘言:
: 3 2
: 2. a > 0, a + a + a = a 求 a (n->∞)
: 1 n+1 n+1 n+1 n n
: 我知道在n->∞時 可以用a_n = a_(n+1) = 固定值α 解出未知數,
: 但最後解出α=0或-1時,如何得知-1不合?
(a_(n+1))^3 + (a_(n+1))^2 + a_(n+1) = (a_(n+1))( (a_(n+1))^2 + a_(n+1) + 1)
又(a_(n+1))^2 + a_(n+1) + 1 恆 > 0 因判別式 < 0,領導係數 > 0
a_n
=> a_(n+1) = ------------------------------ ≧ 0 by induction on n
a_(n+1))^2 + a_(n+1) + 1
且
a_n a_n
a_(n+1) = ------------------------------ < --------- = a_n
a_(n+1))^2 + a_(n+1) + 1 1
此數列遞減有下界,由實數完備性知極限存在,此時用令
lim a_n = lim a_(n+1) = α = lim (a_(n+1))^3 + (a_(n+1))^2 + a_(n+1)
n→∞ n→∞ n→∞
= α^3 + α^2 + α
解出α = 0 或 -1 但負明顯不合 by a_n ≧ 0 for all n
: 4
: 3. 類似的問題, a1 = 2(1+√5) , a_(n+1) = -------
: a_n - 2
: 求a_n極限值,
: 得到 α=1±√5 如何得知正數不合?
: 謝謝:)
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