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Sequence

∀n∈N\forall n\in \mathbb{N} we define the list number …,xn,xn+1,…\dots,x_n,x_{n+1},\dots then {xn}\{x_n\} is a sequence, or (xn)(x_n); some time we have multiple variable and we may have something like (xa,b)a∈N(x_{a,b})_{a\in N} means this sequence focus on the changing of aa

Supreme: we denote supreme as ∀i,L≥ri  ⟹  sup⁡(ri)=L\forall i, L \ge r_i \implies \sup(r_i) = L

Infimum: we denote infimum as ∀i,L≤ri  ⟹  inf⁡(ri)=L\forall i, L \le r_i \implies \inf(r_i) = L

Limit of Sequence & Cauchy Sequence​

  1. We say a sequence {an}n=1∞\{a_n\}^{\infty}_{n=1} has a limit LL or it's convergent to LL if ∀ϵ>0\forall \epsilon>0, ∣an−L∣<ϵ|a_n-L|<\epsilon. Equivalent, lim⁡n→∞an=L\lim_{n\to \infty}a_n = L
    • The limit of a convergent sequence is unique
  2. If lim⁡n→∞an=L\lim_{n\to \infty}a_n = L, lim⁡n→∞bn=M\lim_{n\to \infty}b_n = M and α∈R\alpha\in\mathbb{R} then:
    • lim⁡n→∞an+bn=L+M\lim_{n\to \infty}a_n +b_n = L + M
    • lim⁡n→∞αan=αL\lim_{n\to \infty}\alpha a_n = \alpha L
    • lim⁡n→∞anbn=LM\lim_{n\to \infty}a_nb_n = LM
    • lim⁡n→∞anbn=LM\lim_{n\to \infty}\frac{a_n}{b_n} = \frac{L}{M} if M≠0M\ne0
  3. Squeeze Theorem: if three sequence {an},{bn},{cn}\{a_n\},\{b_n\},\{c_n\} satisfy an≤bn≤cn,∀n>1a_n\le b_n\le c_n,\forall n>1 and lim⁡n→∞an=lim⁡n→∞cn=L\lim_{n\to \infty}a_n = \lim_{n\to \infty}c_n =L, then lim⁡n→∞bn=L\lim_{n\to \infty}b_n = L
  4. If a sequence {an}n=1∞\{a_n\}^{\infty}_{n=1} is converging to LL, then ∀ϵ>0,∃N∈N,∣am−an∣<ϵ\forall \epsilon>0,\exists N\in\mathbb{N}, |a_m-a_n|<\epsilon for all m,n≥Nm,n\ge N
  5. If a sequence of R\mathbb{R} satisfied ∀ϵ>0,∃N∈N,∣am−an∣<ϵ\forall \epsilon>0,\exists N\in\mathbb{N}, |a_m-a_n|<\epsilon for all m,n≥Nm,n\ge N then it is called Cauchy Sequence.
    • Every Cauchy Sequence is bounded (has a supreme and a infimum)
    • A sequence converges   ⟺  \iff Cauchy Sequence
  6. A subset SS of R\mathbb{R} is a complete set if every Cauchy Sequence in SS converges to a point in SS
  7. ∀y∈R\forall y \in\mathbb{R}, ∃\exists a sequence of Q\mathbb{Q} that converges to yy

Monotone Sequence​

A sequence {an}n≥1\{a_n\}_{n\ge1}, if an≤an+1a_n\le a_{n+1}, then it's increasing. If an≥an+1a_n\ge a_{n+1}, then it's decreasing. A sequence either increase or decrease is a monotone sequence.

  • A not monotone sequence example {an}n≥1\{a_n\}_{n\ge1}, where an=(−1)nna_n = (-1)^nn

Monotone Convergence Theorem:

  • A monotone increasing sequence and bounded above converges to its supremum.
  • A monotone decreasing sequence and bounded below converges to its infimum.

Subsequence​

A subsequence of a sequence {an}n=1∞\{a_n\}^{\infty}_{n=1} is a sequence {ank}n=1∞=an1,an2,an3,…\{a_{n_k}\}^{\infty}_{n=1} = {a_{n_1},a_{n_2}},a_{n_3},\dots where n1,n2,n3,…n_1,n_2,n_3,\dots

Bolzano-Weierstrass Theorem: Every bounded sequence of real numbers has a convergent subsequence

infinite Series​

An infinite series is an infinite sum ∑n=1∞an=a1+a2+a3+…\sum_{n=1}^{\infty}a_n=a_1+a_2+a_3+\dots

  • Partial sums of a infinite series is a sequence. E.g. Sn=∑j=1najS_n = \sum_{j=1}^{n}a_j
  • ∑n=1∞an\sum_{n=1}^{\infty}a_n converges   ⟺  \iff lim⁡n→∞Sn=L\lim_{n\to \infty}S_n = L
  • ∑n=1∞an<∞  ⟹  \sum_{n=1}^{\infty}a_n<\infty\implies ∑n=1∞an\sum_{n=1}^{\infty}a_n converge
  • ∑n=1∞an\sum_{n=1}^{\infty}a_n converges   ⟺  \iff ∀ϵ>0,∃N∈N,n>m≥N,∣am+1+⋯+an∣<ϵ\forall \epsilon>0,\exists N\in\mathbb{N},n>m\ge N, |a_{m+1}+\dots+a_n|<\epsilon
  • If ∑n=1∞an\sum_{n=1}^{\infty}a_n converges, lim⁡n→∞an=0\lim_{n\to \infty}a_n = 0
  • ∑n=1∞an\sum_{n=1}^{\infty}a_n absolutely converges   ⟺  \iff ∑n=1∞∣an∣\sum_{n=1}^{\infty}|a_n| converges
  • ∑n=1∞an\sum_{n=1}^{\infty}a_n conditional converges   ⟺  \iff ∑n=1∞∣an∣\sum_{n=1}^{\infty}|a_n| diverges

Some convergent Test

  • P test
  • Comparison Test
  • Ratio Test
  • Root Test
  • Cauchy Condensation Test
  • Alternating Series Test
  • Absolute Convergence Test