Basic Concepts of Sequences

Sequences act as functions on positive integers, bridging discrete steps to the continuous ideas of limits and calculus.

What Is a Sequence?

Sequence

A sequence is a function whose independent variable is a positive integer. For each $n \in \mathbb{N}$, the value is denoted $a_n$, and the whole sequence is written $\{a_n\}$.

Function view

$$a : \mathbb{N} \to \mathbb{R}, \quad a(n) = a_n$$

Typical Examples

• Natural numbers: $a_n = n$ (linear growth)
• Harmonic: $a_n = \dfrac{1}{n}$ (used in limits/estimates)
• Alternating: $a_n = (-1)^n$ (sign flips with period $2$)

Ways to Represent Sequences

1. General term: give $a_n$ directly. Example $a_n = 2n + 1$ ⇒ $3, 5, 7, \dots$
2. Recurrence: generate $a_{n+1}$ from earlier terms. Example $a_1 = 1,\; a_{n+1} = 2a_n$ ⇒ $1, 2, 4, 8, \dots$
3. Listing: write early terms for intuition, e.g., $1, \tfrac12, \tfrac13, \dots$

Link between general term and recurrence

If a recurrence $a_{n+1} = f(a_n)$ can be iterated, the closed form may be found by accumulating the relation. Conversely, a general term can often yield a recurrence by rearrangement.

Common Types

Arithmetic sequences

Consecutive terms differ by a constant $d$.

Arithmetic general term

$$a_n = a_1 + (n-1)d$$

Example: $2, 5, 8, 11, \dots$ with $d = 3$. If $d>0$ the sequence increases; if $d<0$ it decreases.

Geometric sequences

Consecutive terms have a constant ratio $q$.

Geometric general term

$$a_n = a_1 \cdot q^{\,n-1}$$

Example: $3, 6, 12, 24, \dots$ with $q = 2$. When $|q| < 1$, $a_n \to 0$, a classic convergence case.

Basic Properties

Boundedness

Bounded sequence

A sequence $\{a_n\}$ is bounded if $\exists M>0$ with $|a_n| \le M$ for all $n$.

Example: $\left\{\tfrac{1}{n}\right\}$ is bounded; $\{n\}$ is unbounded.

Monotonicity

• Increasing: $a_{n+1} \ge a_n$ for all $n$
• Decreasing: $a_{n+1} \le a_n$ for all $n$

A sequence that is both monotonic and bounded converges—a key bridge to limits.

Periodicity and Parity

• Periodic: $\exists T \in \mathbb{N}$ with $a_{n+T} = a_n$ (e.g., $(-1)^n$ with $T=2$).
• Even/odd patterns: extending $\{a_n\}$ to a function can mirror function parity concepts.

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练习题

练习 1

Classify the sequence and give the parameter:

1. $a_n = 4 - 3n$
2. $b_n = 5 \cdot \left(-\dfrac{1}{2}\right)^{n-1}$

参考答案

1. $a_{n+1} - a_n = -3$ ⇒ arithmetic, $d = -3$.
2. $\dfrac{b_{n+1}}{b_n} = -\tfrac12$ ⇒ geometric, $q = -\tfrac12$.

练习 2

Sequence $\{c_n\}$ satisfies $c_1 = 2$, $c_{n+1} = c_n + 2^{n}$. Find a closed form.

参考答案

Sum the recurrence: $c_{n+1} = 2 + \sum_{k=1}^{n} 2^{k} = 2 + (2^{n+1}-2) = 2^{n+1}$, so $c_n = 2^{n}$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\mathbb{N}$	数学符号	natural numbers	索引的取值集合
$\mathbb{R}$	数学符号	Real numbers	项的取值集合
$\{a_n\}$	数列记号	sequence a_n	一个数列
$a_n$	元素符号	a sub n	第 $n$ 项
$d$	参数	difference	等差数列公差
$q$	参数	quotient	等比数列公比

中英对照

中文术语	英文术语	音标	说明
数列	sequence	/ˈsiːkwəns/	以正整数为索引的函数
通项公式	general term	/ˈdʒenərəl tɜːrm/	直接给出第 $n$ 项
递推公式	recurrence relation	/rɪˈkɜːrəns rɪˈleɪʃən/	通过前项生成后项
等差数列	arithmetic sequence	/ˌærɪθˈmetɪk ˈsiːkwəns/	相邻项差为常数
等比数列	geometric sequence	/ˌdʒiːəˈmetrɪk ˈsiːkwəns/	相邻项比为常数
有界	bounded	/ˈbaʊndɪd/	绝对值被同一常数约束
单调	monotonic	/ˌmɒnəˈtɒnɪk/	项随索引一致增或减

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