Week 1 - Sequences

1. Definition

  • A sequence can be thought of as a list of numbers written in a definite order:
    • a1,a2,a3,a4,,an,a_1, a_2, a_3, a_4, \ldots , a_n, \ldots

1.1. Notation

  • The sequence a1,a2,a3,{a_1, a_2, a_3, \ldots} also denoted by
    • {an}\{a_n\} or {an}n=1\{a_n\}_{n=1}^{\infty}
      • nNn \in \mathbb{N} (whole number)

1.2. Example: The Fibonacci Sequence

  • Definition: {fn}\{f_n\} is defined recursively by the conditions
    • f1=1f_1 = 1, f2=1f_2 = 1, fn=fn1+fn2f_n = f_{n-1} + f_{n-2}, n3n \ge 3

1.3. Different Ways to Present Sequence

  • Two sequences ana_n and bnb_n are equal if they begin at the same index N, and an=bna_n = b_n whenever nNn \ge N.
  • For example:
    • an=2n for n0a_n = 2^n\ \text{for}\ n \ge 0
    • b0=1 and bn=2bn1b_0 = 1\ \text{and}\ b_n = 2 \cdot b_{n-1}

2. Examples

2.1. Tribonacci Sequence

  • Definition: a0=a1=a2=1a_0 = a_1 = a_2 = 1, an=an1+an2+an3a_n = a_{n-1} + a_{n-2} + a_{n-3}
    • Samples: 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355
  • We can build a new sequence from this:
    • bn=an+1anb_n = \frac{a_n + 1}{a_n}
    • Samples: 1,1,1,3,53,95,179,3117,5731,10557,193105,3551931, 1, 1, 3, \frac{5}{3}, \frac{9}{5}, \frac{17}{9}, \frac{31}{17}, \frac{57}{31}, \frac{105}{57}, \frac{193}{105}, \frac{355}{193}
    • So limnbn=?\displaystyle\lim_{n \to \infty} b_n = ?

2.2. Arithmetic Progression

  • Definition: An arithmetic progression is an sequence with a common difference between the terms.
  • Example: 5,12,19,26,33,:an=5+7n5, 12, 19, 26, 33, \ldots : a_n = 5 + 7n
  • Formula: an=a0+dna_n = a_0 + d_n
  • In a arithmetic progression, each term is the arithmetic mean of its neighbors.
    • arithmetic mean: an=an1+an+12\displaystyle a_n = \frac{a_{n-1} + a_{n+1}}{2}

2.3. Geometric Progression

  • Definition: An geometric progression is an sequence with a common ratio between the terms.
  • Example: 3,6,12,24,48,96,:an=32n3, 6, 12, 24, 48, 96, \ldots : a_n = 3 \cdot 2^n
  • Formula: an=a0rna_n = a_0 \cdot r^n
  • In a geometric progression, each term is the geometric mean of its neighbors.
    • geometric mean: an=an1an+1\displaystyle a_n = \sqrt{a_{n-1} a_{n+1}}
      • for example, an area of a square with side length of ab\sqrt{ab} is abab
  • limnan\displaystyle\lim_{n \to \infty} a_n
    • if the common ratio > 1, then limnan=\displaystyle\lim_{n \to \infty} a_n = \infty
    • if the common ratio < 1, then limnan=0\displaystyle\lim_{n \to \infty} a_n = 0

3. Limit of a Sequence

  • Definition: limnan=L\displaystyle \lim_{n \to \infty} a_n = L means that, for every ϵ>0\epsilon > 0, there is a whole number N, so that, whenever nNn \ge N, anL<ϵ\lvert a_n - L \rvert < \epsilon.

4. Sequence Bounded

  • Definition:
    • ana_n is "bounded above" means there is a real number M, so that, for all n0,anMn \ge 0, a_n \le M.
    • ana_n is "bounded below" means there is a real number M, so that, for all n0,anMn \ge 0, a_n \ge M.
    • ana_n is "bounded" means ana_n is "bounded above" and "bounded below".
  • Example:
    • an=sinn,1sinn1a_n = \sin n, -1 \le \sin n \le 1 . So ana_n bounded.
    • bn=nsin(πnn)b_n = n \cdot \sin(\frac{\pi \cdot n}{n}), not bounded.

5. Sequence Increasing

  • Definition:
    • A sequence (ana_n) is increasing if whenever m > n, then am>ana_m > a_n.
    • A sequence (ana_n) is decreasing if whenever m > n, then am<ana_m < a_n.
    • A sequence (ana_n) is non-decreasing if whenever m > n, then amana_m \ge a_n.
    • A sequence (ana_n) is non-increasing if whenever m > n, then amana_m \le a_n.

6. The Monotone Convergence Theorem

  • Definition: If the sequence (ana_n) is bounded and monotone, then limnan\displaystyle\lim_{n \to \infty} a_n exists.
  • Example: a1=1,an+1=an+2a_1 = 1, a_{n+1} = \sqrt{a_n +2}
    • To prove the limit of this sequence exists, we need to this sequence is
      • bounded: 0an20 \le a_n \le 2
        • 0an+1=2+2=20 \le a_{n+1} = \sqrt{2+2} = 2
        • 0a120a220 \le a_1 \le 2 \implies 0 \le a_2 \le 2 \implies \ldots
      • monotone(non-decreasing): anan+1a_n \le a_{n+1}
        • anan+1=an+2a_n \le a_{n+1} = \sqrt{a_n + 2}
        • an2an20a_n^2 - a_n - 2 \ge 0
        • (2an)(1+an)0(2 - a_n)(1 + a_n) \ge 0 which is true
    • So the limit of ana_n exists.

7. Extra

7.1. A Sequence Includes Every Integer

  • Cn={(n+1)/2if n oddn/2if n evenC_n = \begin{cases} -(n+1)/2 &\text{if } n\ \text{odd} \\ n/2 &\text{if } n\ \text{even} \end{cases}
    • Starting with index 0.
  • An infinite quantity is a quantity that won't be smaller, when you take something away.
    • Like we take away the negative integers from all integers, which is still infinite.
  • Note: I've taken a similar course talked about it:

7.2. A Sequence Includes Every Real Number

8. Words

  • monotone function 单调函数;单弹数
  • monotone increasing 单调递增
  • monotone regression 单调回归
  • parity n. 平价;同等;相等
  • quantitative adj. 定量的;量的,数量的
  • qualitative adj. 定性的;质的,性质上的
  • quantitative and qualitative change 量变与质变

results matching ""

    No results matching ""