Week 1 - Sequences #

Definition #

A sequence can be thought of as a list of numbers written in a definite order:
- $a_1, a_2, a_3, a_4, \ldots , a_n, \ldots$

The sequence ${a_1, a_2, a_3, \ldots}$ also denoted by
- ${a_n}$ or ${a_n}_{n=1}^{\infty}$
  - $n \in \mathbb{N}$ (whole number)

Definition: ${f_n}$ is defined recursively by the conditions
- $f_1 = 1$, $f_2 = 1$, $f_n = f_{n-1} + f_{n-2}$, $n \ge 3$

Two sequences $a_n$ and $b_n$ are equal if they begin at the same index N, and $a_n = b_n$ whenever $n \ge N$.
For example:
- $a_n = 2^n\\ \text{for}\\ n \ge 0$
- $b_0 = 1\\ \text{and}\\ b_n = 2 \cdot b_{n-1}$

Definition: $a_0 = a_1 = a_2 = 1$, $a_n = a_{n-1} + a_{n-2} + a_{n-3}$
- Samples: 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355
We can build a new sequence from this:
- $b_n = \frac{a_n + 1}{a_n}$
- Samples: $1, 1, 1, 3, \frac{5}{3}, \frac{9}{5}, \frac{17}{9}, \frac{31}{17}, \frac{57}{31}, \frac{105}{57}, \frac{193}{105}, \frac{355}{193}$
- So $\displaystyle\lim_{n \to \infty} b_n = ?$

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

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

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

Definition:
- $a_n$ is “bounded above” means there is a real number M, so that, for all $n \ge 0, a_n \le M$.
- $a_n$ is “bounded below” means there is a real number M, so that, for all $n \ge 0, a_n \ge M$.
- $a_n$ is “bounded” means $a_n$ is “bounded above” and “bounded below”.
Example:
- $a_n = \sin n, -1 \le \sin n \le 1$. So $a_n$ bounded.
- $b_n = n \cdot \sin(\frac{\pi \cdot n}{n})$, not bounded.

Definition:
- A sequence ($a_n$) is increasing if whenever m > n, then $a_m > a_n$.
- A sequence ($a_n$) is decreasing if whenever m > n, then $a_m < a_n$.
- A sequence ($a_n$) is non-decreasing if whenever m > n, then $a_m \ge a_n$.
- A sequence ($a_n$) is non-increasing if whenever m > n, then $a_m \le a_n$.

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

$$C_n = \begin{cases} -(n+1)/2 &\text{if } n\\ \text{odd} \\ n/2 &\text{if } n\\ \text{even} \end{cases}$$
- 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:
- Effective Thinking Through Mathematics (Week 4: Telling the Story of Infinity)

Real Number: The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356…, the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as π (3.14159265…). – from Wikipedia
Doesn’t exist.
Prove: Effective Thinking Through Mathematics (Week 4#infinity-comes-in-different-sizes)