序列(Sequence)

單調序列 (Monotonic sequences)

  • {sn} \lbrace s_n \rbrace 為實數序列。
  • 單調遞增序列(increasing sequence): snsn+1s_n \leq s_{n+1}, nN\forall n \in \mathbb{N}.
  • 單調遞減序列(decreasing seqeunce): snsn+1s_n \geq s_{n+1}, nN\forall n \in \mathbb{N}.

收斂序列 (Convergence sequences)

  • A sequence {pn}\lbrace p_n \rbrace in metric space (X,d)(X, d) is said to be converge in XX if there is a point pXp \in X such that:

    • ϵ>0\forall \epsilon > 0  n0N\exists \ n_0 \in \mathbb{N}  d(pn,p)<ϵ\ni \ d(p_n, p) < \epsilon, nn0\forall n \geq n_0.

    • 通常寫為 limnpn=p\lim_{n \rightarrow \infty} p_n = p or pnpp_n \rightarrow p.

    • 收斂可解釋為只要nn夠大,sequence與收斂值之間的差距可以任意小。

  • If {pn}\lbrace p_n \rbrace does not converge, it is said to diverge (發散) .

  • Sequence是否收斂,除了自身的特性外,還必須確認收斂點pp是否為metric space中的元素。

    • E.g. {1/n}\lbrace 1/n \rbraceR\mathbb{R}會收斂到0,但在R+\mathbb{R}^+, with metric ρ(x,y)=xy\rho(x,y)=\vert x-y \vert不收斂,因0R+ 0 \notin \mathbb{R}^+ .
  • Theorem: Let {pn}\lbrace p_n \rbrace be a sequence in a metric space XX., then

    • limnpn=pX\lim_{n \rightarrow \infty} p_n = p \in X \Leftrightarrow every neighborhood of pp contains all but finitely many of the terms of {pn}\lbrace p_n \rbrace.

    • (收斂的唯一性) Let p1,p2Xp_1, p_2 \in X, limnpn=p1\lim_{n \rightarrow \infty} p_n = p_1 and limnpn=p2\lim_{n \rightarrow \infty} p_n = p_2 then p1=p2p_1 = p_2.

    • (收斂序列必有界) If {pn}\lbrace p_n \rbrace converges, then {pn}\lbrace p_n \rbrace bounded.

    • If EXE \subset X and pp is a limit point of EE then there is a sequence {pn}\lbrace p_n \rbrace in EE such that limnpn=p\lim_{n \rightarrow \infty} p_n = p.

Sequence 運算

  • Let {sn}\lbrace s_n \rbrace and {tn}\lbrace t_n \rbrace are complex sequences, limnsn=s\lim_{n \rightarrow \infty} s_n = s and limntn=t\lim_{n \rightarrow \infty} t_n = t then

    • limn(sn+tn)=s+t\lim_{n \rightarrow \infty} (s_n + t_n ) = s+t.

    • cC\forall c \in \mathbb{C}, limncsn=cs\lim_{n \rightarrow \infty} cs_n = cs, limn(c+sn)=c+s\lim_{n \rightarrow \infty} (c + s_n) = c + s.

    • limnsntn=st\lim_{n \rightarrow \infty} s_n t_n = st.

    • limn1n=1s\lim_{n \rightarrow \infty} \frac{1}{n} = \frac{1}{s} if sn0 ns_n \neq 0 \ \forall n and s0s \neq 0.

Subsequence

  • Subsequence為sequence中不連續的元素(但順序不變)所形成的序列,與演算法中subsequence定義相同。

  • Definition

    • Given a sequence {pn}\lbrace p_n \rbrace, consider a sequence {nk}\lbrace n_k \rbrace of positive integers such that n1<n2<<nk< n_1 < n_2 < \cdots < n_k < \cdots then the sequence {pnk}\lbrace p_{n_k} \rbrace is called subsequence of {pn}\lbrace p_n \rbrace.
  • If limnpn=p\lim_{n \rightarrow \infty} p_n = p \Leftrightarrow every subsequence of {pn}\lbrace p_n \rbrace converges to pp.

  • Theorem: If {pn}\lbrace p_n \rbrace is a sequence in a compact metric space (X,d)(X, d), then some subsequence of {pn}\lbrace p_n \rbrace converges to a point of XX.

Cauchy sequence

  • Definition

    • A sequence {pn}\lbrace p_n \rbrace in m a metric space (X,d)(X, d) is said to be a Cauchy sequence if ϵ>0\forall \epsilon > 0 n0N\exists n_0 \in \mathbb{N} d(d(pn,pm)<ϵ\ni d(d(p_n, p_m) < \epsilon, n,m>n0\forall n, m > n_0.

    • Cauchy sequence可用於判定sequence的收斂性,而不需要知道sequence收斂之值為何。

  • E.g. Rn n\mathbb{R}^n \ \forall n are complete metric spaces, 所有的Cauchy sequences均收斂。

    • Converge sequence \Rightarrow Cauchy sequence。反之不一定成立,必須要加上compact metric space XX的條件以保證收斂值位於metric space中。
  • Properties:

    • EXE \subset X is called bounded set if diamE<diamE < \infty.

    • If {pn}\lbrace p_n \rbrace is a sequence in XX and ENE_N consists of the points pN,pN+1,pN+2,p_N, p_{N+1}, p_{N+2}, \cdots, then {pn}\lbrace p_n \rbrace is a Cauchy sequence \Leftrightarrow limNdiamEN=0\lim_{N \rightarrow \infty} diamE_N = 0.

    • If E¯\bar{E} is the closure of a set EXE \subset X, then diamE¯=diamEdiam\bar{E} = diamE.

    • If knk_n is a sequence of compact sets in XX, knkn+1k_n \supset k_{n+1} nN\forall n \in \mathbb{N}, and if limndiamkn=0\lim_{n \rightarrow \infty} diam k_n =0, then n=1kn\cap_{n=1}^{\infty}k_n conststs of exactly one point.

發散序列(Divergence sequence)

    • 定義: 發散序列( Divergence sequence)
    • {sn}\lbrace s_n \rbrace 為實數序列且有以下性質:
    • (往正無窮大發散) MR\forall M \in \mathbb{R} n0N\exists n_0 \in \mathbb{N} snM\ni s_n \geq M nn0\forall n \geq n_0. usually denoted as limnsn+\lim_{n \rightarrow \infty} s_n \rightarrow +\infty.
    • (往負無窮大發散) MR\forall M \in \mathbb{R} n0N\exists n_0 \in \mathbb{N} snM\ni s_n \leq M nn0\forall n \geq n_0, usually denoted as limnsn\lim_{n \rightarrow \infty} s_n \rightarrow -\infty.

序列的上極限

    • 定義: 序列的上極限(upper limit of sequence)
    • 令序列 {sn}\lbrace s_n \rbrace 有上界且 Mn=sup{sn,sn+1,sn+2,,},M_n = \sup \lbrace s_n, s_{n+1}, s_{n+2}, \cdots,\rbrace,MnMn+1Mn+2,M_n \geq M_{n+1} \geq M_{n+2}, \cdots .
    • 若序列 MnM_n 收斂, 則序列 {sn}\lbrace s_n \rbrace 的上極限寫為 limsupnsn=limnMn\lim \sup_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} M_n
    • .
    • 若序列 MnM_n 發散, 則序列的上極限也發散, limsupnsn=\lim \sup_{n \rightarrow \infty} s_n = \infty.
    • 若序列 {sn}\lbrace s_n \rbrace 無上界,則序列的上極限發散, limsupnsn=\lim \sup_{n \rightarrow \infty} s_n = \infty.
    • limsupnsn=limn(supmnsm)=infn0supknsk=inf{sup{skkn}n0} \lim \sup_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} \left( \sup_{m \geq n} s_m \right) = \inf_{n \geq 0} \sup_{ k \geq n} s_k = \inf \{ \sup\{ s_k| k \geq n \} | n \geq 0 \}

    • limsupnsn=liminfnsn \lim \sup_{n \rightarrow \infty} s_n = - \lim \inf_{n \rightarrow \infty} - s_n .
  • MM為序列{sn}\{s_n\}的上極限,則必須滿足以下兩個條件:

    • ϵ>0 n0Nsn<M+ϵ nn0\forall \epsilon > 0\ \exists n_0 \in \mathbb{N} \ni s_n < M + \epsilon \ \forall n \geq n_0 .
      • 此條件為序列極限有上界,但不唯一。
      • a,bR, a<bϵ>0, a<b+ϵa,b \in \mathbb{R},\ a < b \Leftrightarrow \forall \epsilon > 0,\ a < b+ \epsilon.
    • ϵ>0 and n1Nsn>Mϵ nn1 \exists \epsilon > 0 \text{ and } n_1 \in \mathbb{N} \ni s_n > M - \epsilon \ \forall n \geq n_1 .
      • 此條件為序列中有無窮多個元素(但非全部)落於MϵM-\epsilon的右側。
    • 滿足以上兩個條件時,寫為limsupns=M\lim \sup_{n \rightarrow \infty} s_ = M .
序列的上極限與下極限。
  • E.g. {sn}={(1)n}\lbrace s_n \rbrace = \lbrace (-1)^n \rbracenN\forall n \in \mathbb{N},可知{sn}\lbrace s_n \rbrace1-111間振盪,所以limsupsn=1\lim \sup s_n = 1, and liminfsn=1\lim \inf s_n = -1.

  • E.g. {sn}={n}\lbrace s_n \rbrace = \lbrace -n \rbrace is bounded above, and nN\forall n \in \mathbb{N}, Mn=nM_n = -n, limsupnsn=\therefore \lim \sup_{n \rightarrow \infty} s_n = -\infty.

數列的下極限

    • 定義: 序列的下極限(lower limit of sequence)
  • 令新列 {sn}\lbrace s_n \rbrace 有下界, 且 mn=inf{sn,sn+1,sn+2,,}m_n = \inf \lbrace s_n, s_{n+1}, s_{n+2}, \cdots, \rbracemnmn+1mn+2,m_n \leq m_{n+1} \leq m_{n+2}, \cdots .
  • 若序列 mnm_n 收斂,則序列 sns_n 的下極限寫為 liminfnsn=limnmn\lim \inf_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} m_n.
  • 若序列 mnm_n 發散,則序列的下極限也發散, liminfnsn=\lim \inf_{n \rightarrow \infty} s_n = \infty.
  • 若序列 {sn}\lbrace s_n \rbrace 無下界, 則序列的下極限也發散 liminfnsn=\lim \inf_{n \rightarrow \infty} s_n = -\infty.
  • liminfnsn=limn(infmnsm)=supn0infknsk=sup{inf{skkn}n0} \lim \inf_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} \left( \inf_{m \geq n} s_m \right) = \sup_{n \geq 0} \inf_{k \geq n} s_k = \sup\{ \inf\{ s_k| k \geq n \} | n \geq 0 \}
  • </ul></div>

    • E.g. liminfn(1)n=1 \lim \inf_{n \rightarrow \infty }(-1)^n = -1.

    • E.g. liminfnn=\lim \inf_{n \rightarrow \infty} n = \infty.

    • E.g. liminfn(n)=\lim \inf_{n \rightarrow \infty} (-n) = -\infty.

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