Week 1 - Sequences
- A sequence can be thought of as a list of numbers written in a definite order:
- The sequence also denoted by
- (whole number)
1.2. Example: The Fibonacci Sequence
- Definition: is defined recursively by the conditions
- , , ,
1.3. Different Ways to Present Sequence
- Two sequences and are equal if they begin at the same index N, and whenever .
- For example:
2.1. Tribonacci Sequence
- Definition: ,
- Samples: 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355
- We can build a new sequence from this:
2.2. Arithmetic Progression
- Definition: An arithmetic progression is an sequence with a common difference between the terms.
- In a arithmetic progression, each term is the arithmetic mean of its neighbors.
- arithmetic mean:
2.3. Geometric Progression
- Definition: An geometric progression is an sequence with a common ratio between the terms.
- In a geometric progression, each term is the geometric mean of its neighbors.
- geometric mean:
- for example, an area of a square with side length of is
- if the common ratio > 1, then
- if the common ratio < 1, then
3. Limit of a Sequence
- Definition: means that, for every , there is a whole number N, so that, whenever , .
4. Sequence Bounded
- is "bounded above" means there is a real number M, so that, for all .
- is "bounded below" means there is a real number M, so that, for all .
- is "bounded" means is "bounded above" and "bounded below".
- . So bounded.
- , not bounded.
5. Sequence Increasing
- A sequence () is increasing if whenever m > n, then .
- A sequence () is decreasing if whenever m > n, then .
- A sequence () is non-decreasing if whenever m > n, then .
- A sequence () is non-increasing if whenever m > n, then .
6. The Monotone Convergence Theorem
- Definition: If the sequence () is bounded and monotone, then exists.
- To prove the limit of this sequence exists, we need to this sequence is
- which is true
- So the limit of exists.
7.1. A Sequence Includes Every Integer
- 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
- 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