Introduction to sequence

• Definition of a Sequence
  • Example:
• Notation and Examples of Sequences
  • Notation:
  • Examples:
    • 1. Constant Sequence:
    • 2. Arithmetic Sequence:
    • 3. Geometric Sequence:
    • 4. Harmonic Sequence:
• Convergence and Divergence of Sequences
  • Convergence:
  • Divergence:
  • Example of Convergence:
  • Example of Divergence:
• Limits of Sequences
  • Formal Definition:
  • Limit Laws for Sequences:
• Bounded and Monotonic Sequences
  • Bounded Sequences:
    • Example of a Bounded Sequence:
  • Monotonic Sequences:
    • Example of a Monotonically Increasing Sequence:
    • Example of a Monotonically Decreasing Sequence:
  • The Monotone Convergence Theorem:
    • Example:

Definition of a Sequence

• A sequence is an ordered list of numbers, often defined as a function whose domain is the set of natural numbers. Each number in the list is called a term of the sequence.

• Formally, a sequence is a function $f:\mathbb{N}\to \mathbb{R}$, where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. The value of the function $f$ at a natural number $n$ is called the $n$-th term of the sequence and is typically denoted by $a_n$.

Example:

If $a_n=\frac{1}{n}$, the sequence is:

This gives the sequence:

Notation and Examples of Sequences

Notation:

A sequence is typically written in one of the following forms:

• $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$: A sequence where the $n$-th term is $a_n$.
• $(a_n{)}_{n=1}^{\mathrm{\infty }}$: Another common notation for sequences.

The general term of the sequence is given by a formula $a_n$, which determines the value of each term based on $n$.

Examples:

1. Constant Sequence:

A sequence where every term is the same, such as $a_n=c$ for all $n\in \mathbb{N}$.

• Example: $a_n=5$, giving the sequence $(5,5,5,5,\dots )$.

2. Arithmetic Sequence:

A sequence where each term is the previous term plus a constant difference $d$.

• Example: $a_n=2n$, giving the sequence $(2,4,6,8,\dots )$.

3. Geometric Sequence:

A sequence where each term is the previous term multiplied by a constant ratio $r$.

• Example: $a_n=3\cdot 2^n$, giving the sequence $(6,12,24,48,\dots )$.

4. Harmonic Sequence:

A sequence where each term is the reciprocal of an arithmetic sequence.

• Example: $a_n=\frac{1}{n}$, giving the sequence $(1,\frac{1}{2},\frac{1}{3},\dots )$.

Convergence and Divergence of Sequences

Convergence:

A sequence $\{a_n\}$ converges to a real number $L$ if, as $n$ becomes very large, the terms of the sequence get arbitrarily close to $L$. More formally, $\{a_n\}$ converges to $L$ if for every $ϵ>0$, there exists a natural number $N$ such that for all $n>N$, we have:

In this case, we write:

Divergence:

If a sequence does not converge to a finite limit, we say that it diverges. A sequence can diverge in various ways:

• Divergence to infinity: The terms of the sequence grow without bound as $n$ increases.

  • Example: The sequence $a_n=n$ diverges to infinity as $n\to \mathrm{\infty }$.

• Oscillatory divergence: The terms of the sequence oscillate between values without approaching a single limit.

  • Example: The sequence $a_n=(-1{)}^n$ oscillates between $1$ and $-1$ and does not converge.

Example of Convergence:

The sequence $a_n=\frac{1}{n}$ converges to $0$:

Example of Divergence:

The sequence $a_n=n$ diverges to infinity:

Limits of Sequences

The limit of a sequence describes the value (if any) that the terms of the sequence approach as $n$ increases without bound.

Formal Definition:

The sequence $\{a_n\}$ has a limit $L$ if for every $ϵ>0$, there exists an integer $N$ such that for all $n>N$:

This definition formalizes the idea that beyond some index $N$, the terms of the sequence are all within a small distance $ϵ$ of $L$.

Limit Laws for Sequences:

If $\underset{n\to \mathrm{\infty }}{lim}{a}_n=L$ and $\underset{n\to \mathrm{\infty }}{lim}{b}_n=M$, then:

1. Sum Rule: $\underset{n\to \mathrm{\infty }}{lim}(a_n+b_n)=L+M$
2. Difference Rule: $\underset{n\to \mathrm{\infty }}{lim}(a_n-b_n)=L-M$
3. Product Rule: $\underset{n\to \mathrm{\infty }}{lim}(a_n\cdot b_n)=L\cdot M$
4. Quotient Rule: $\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=\frac{L}{M}$, provided $M\ne 0$

Bounded and Monotonic Sequences

Bounded Sequences:

A sequence $\{a_n\}$ is called bounded if there exists a real number $M$ such that for all $n$, $|a_n|\le M$. In other words, the terms of the sequence stay within a fixed range.

• A sequence is bounded above if there exists an $M$ such that $a_n\le M$ for all $n$.
• A sequence is bounded below if there exists an $m$ such that $a_n\ge m$ for all $n$.

Example of a Bounded Sequence:

The sequence $a_n=\frac{1}{n}$ is bounded since $0\le a_n\le 1$ for all $n$.

Monotonic Sequences:

A sequence $\{a_n\}$ is monotonic if its terms are always increasing or always decreasing.

• A sequence is monotonically increasing if $a_n\le a_{n+1}$ for all $n$.
• A sequence is monotonically decreasing if $a_n\ge a_{n+1}$ for all $n$.

Example of a Monotonically Increasing Sequence:

The sequence $a_n=n$ is monotonically increasing since $a_n\le a_{n+1}$ for all $n$.

Example of a Monotonically Decreasing Sequence:

The sequence $a_n=\frac{1}{n}$ is monotonically decreasing since $a_n\ge a_{n+1}$ for all $n$.

The Monotone Convergence Theorem:

If a sequence is both monotonic and bounded, then it converges.

Example:

The sequence $a_n=\frac{1}{n}$ is both monotonically decreasing and bounded below by $0$, so by the monotone convergence theorem, it converges to $0$: