Study Notes on Sequences, Series, and Power Series

Definition of Infinite Sequence:
- An infinite sequence, or just a sequence, is a list of numbers written in a definite order.
- The terms of the sequence are denoted as follows:
- $a_1$: first term
- $a_2$: second term
- In general, $a_n$ is the nth term.
- An infinite sequence consists of terms $an$ that have successors denoted by $a{n+1}$.
- Function Representation:
- A sequence can be defined as a function $f$ whose domain is the set of positive integers, where for every positive integer $n$, there is a corresponding number $a_n$.

Standard Notation of Sequences:
- Instead of writing $f(n)$ for the value of the function at number $n$, the sequence is usually expressed in the form:
- $ ext{{Sequence Notation: }} igg an igg{n=1}^{ ext{infinity}}$ (unless stated otherwise, $n$ starts at 1).

Examples of Sequences Defined by Formulas:
- (a) Sequence defined by:
- $a_n = rac{1}{2} $
- Corresponding terms: $1, 1, 1, 1, 1, 1, …$
- (b) Sequence starting from $n=2$:
- Formula: $a_n = n + 1$
- Terms for $n=2, 3, 4, 5, …$: $3, 4, 5, 6, …$
- (c) Sequence starting from $n=1$:
- $a_n = n + 1$
- Terms would be: $2, 3, 4, 5, …$
- (d) Alternate sequence generation:
- Model: $a_n = (-1)^n rac{1}{3^n}$
- Terms: $1, - rac{1}{3}, rac{1}{9}, - rac{1}{27}, …$

Conceptual Understanding:
- A sequence can be visualized by plotting terms on a number line or a graph.
- Example of limits:
- As $n$ approaches infinity, the terms of the sequence approach 1.
- Mathematical Notation:
- We express this concept as:
  ext{lim}{n o ext{∞}} an = 1

Intuitive Definition:
- A sequence ${an}$ has the limit $L$ if we can make the terms $an$ as close to $L$ as we want by selecting a sufficiently large $n$.
- If such a limit exists, the sequence is termed as convergent; otherwise, it is divergent.

A sequence is considered divergent if its terms do not approach a single number.
- Example: Sequence oscillating between two numbers or increasing without bound as $n$ increases.
- Notation for divergence to infinity:
- ext{lim}{n o ext{∞}} an = ext{∞}

Theorem:
- If a limit exists when $n$ is an integer, the following relation holds:
- ext{lim}{n o ext{∞}} an = L
- As $n$ increases, if $x$ approaches $L$, then both sequences converge towards $L$.

Important Limit Laws for Sequences:
1. Sum Law: If $an$ and $bn$ are convergent sequences, then:
  ext{lim}{n o ext{∞}} (an + bn) = La + L_b
2. Difference Law: If $an$ and $bn$ are convergent sequences:
  ext{lim}{n o ext{∞}} (an - bn) = La - L_b
3. Product Law: If $c$ is a constant:
  ext{lim}{n o ext{∞}} (can) = cL_a
4. Quotient Law: If $bn eq 0$ and $ ext{lim}{n o ext{∞}} bn = Lb
  eq 0$:
  ext{lim}{n o ext{∞}} rac{an}{bn} = rac{La}{L_b}
5. Power Law:
  ext{lim}{n o ext{∞}} (an^p) = L_a^p for any real number $p$.

Monotonic Sequences:
- A sequence ${a_n}$ is said to be increasing if:
- an < a{n+1} ext{ for all } n ext{ such that } n
  eq 1
- Decreasing if:
- an > a{n+1} ext{ for all } n ext{ such that } n
  eq 1
- A sequence is monotonic if it is either increasing or decreasing.

Definition of Bounded Sequence:
- A sequence is bounded above if there exists an upper bound $M$ such that:
- an ext{ for all } n ext{ satisfies } an orall n ext{ with } an ext{ and } an ext{ for all } n
- A sequence that is both bounded above and below is termed bounded.
- Example of a bounded sequence: $a_n = rac{1}{n}$

Monotonic Sequence Theorem: Every bounded monotonic sequence is convergent.
- If ${a_n}$ is increasing and bounded above (or decreasing and bounded below), then it converges to some limit $L$.

The results of Example 11 indicate:
- The sequence is convergent if $-1 < r ≤ 1$ and divergent for all other values of r.

What are the steps for determining is a sequence is convergent or divergent?