4.7 Sequences and Series: Glossary
The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.
In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.
| Word | Pronunciation | Part of speech词性 | Related terminology | Chinese meaning | Example sentence 1 | Example sentence 2 |
|---|---|---|---|---|---|---|
| sequence | /ˈsiːkwəns/ | n. | arithmetic sequence 等差数列； geometric sequence 等比数列； Fibonacci sequence 斐波那契数列 |
数列 | The sequence \(a_n = \frac{1}{n}\) tends to zero. 数列 \(a_n = \frac{1}{n}\) 趋于 \(0\)。 |
The sequence \(a_n\) converges. 数列 \(a_n\) 收敛。 |
| series | /ˈsɪəriːz/ | n. | arithmetic series 等差级数； geometric series 几何级数（等比级数）； harmonic series 调和级数； telescoping series 裂项级数； alternating series 交错级数； power series 幂级数； Taylor series 泰勒级数； Maclaurin series 麦克劳林级数 |
级数 | The series \(\sum \frac{1}{n}\) diverges. 级数 \(\sum \frac{1}{n}\) 发散。 |
The series has infinitely many terms. 该级数有无穷多项。 |
| divergence | /daɪˈvɜːdʒəns/ | n. | diverge v. ；divergent adj. | 发散 | Divergence occurs as \(n \to \infty\). 当 \(n \to \infty\) 时发散。 |
The sequence diverges to infinity. 数列发散到无穷。 |
| convergence | /kənˈvɜːdʒəns/ | n. | converge v.；convergent adj.； disk of convergence 收敛圆盘； radius of convergence 收敛半径 |
收敛 | The series \(\sum a_n\) converges. 级数 \(\sum a_n\) 收敛。 |
The sequence converges to \(L\). 数列收敛到 \(L\)。 |
| approach | /əˈprəʊtʃ/ | v./n. | 趋近 | As \(n \to \infty\), the sequence approaches zero. 当 \(n \to \infty\) 时，该数列趋近于 \(0\)。 |
As \(x\) increases, the graph approaches the horizontal axis. 随着 \(x\) 增大，图像逐渐趋近于横轴。 |
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| term | /tɜːm/ | n. | core term 核心术语； general term 通项 |
术语，项 | The expression has three terms. 该表达式有三项。 |
Each term in the series contributes to the sum. 级数中的每一项都贡献总和。 |
| subscript | /ˈsʌbskrɪpt/ | n. | 下标 | The sequence \(a_1\), \(a_2\), \(a_3\) uses subscripts to distinguish terms. 数列 \(a_1\), \(a_2\), \(a_3\) 用下标区分各项。 |
In \(a_n\), the subscript \(n\) indicates the index. 在 \(a_n\) 中，下标 \(n\) 表示索引。 |
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| index | /ˈɪndeks/ | n. | index of summation 求和指标 | 索引，指标 | In the sum \(\sum_{i=1}^{n} a_i\), \(i\) is the index. 在求和 \(\sum_{i=1}^{n} a_i\)中，\(i\) 是指标。 | The index \(n\) runs from \(1\) to infinity. 指标 \(n\) 从 \(1\) 取到无穷大。 |
| consecutive | /kənˈsekjʊtɪv/ | adj. | 连续的 | The difference between consecutive terms in an arithmetic sequence is constant. 等差数列中相邻项的差是常数。 |
Consider two consecutive integers. 考虑两个连续整数。 |
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| expand | /ɪkˈspænd/ | v. | expansion n. (e.g. Binomial expansion) | 展开 | Expand \((1 + x)^2\) in powers of \(x\). 将 \((1 + x)^2\) 按 \(x\) 的幂展开。 | The function \(\frac{1}{1 − x}\) can be expanded into a series. 函数 \(\frac{1}{1 − x}\) 可以展开成级数。 |
| common difference | /ˈkɒmən ˈdɪfrəns/ | n. phr. | common ratio 公比 | 公差 | Arithmetic sequences have constant common difference. 等差数列有常数公差。 |
We calculate the common difference \(d\). 计算公差 \(d\)。 |
| arithmetic mean | /ˌærɪθˈmetɪk miːn/ | n. phr. | geometric mean 几何平均数 | 算术平均数 | Let \(x_1, x_2, \cdots, x_n\) be numbers, then their arithmetic mean is \(\frac{1}{n} \sum_{i=1}^n x_i\). 设 \(x_1, x_2, \cdots, x_n\) 为数，则它们的算术平均数为 \(\frac{1}{n} \sum_{i=1}^n x_i\)。 |
The arithmetic mean of \(2, 4\), and \(6\) is \(4\). \(2, 4, 6\) 的算术平均数是 \(4\)。 |
| ratio test | /ˈreɪʃiəʊ test/ | n. phr. | root test 柯西根值判别法 | 比值判别法 | If the limit is less than \(1\), the series converges by the ratio test. 如果极限小于 \(1\)，则该级数根据比值判别法收敛。 |
By the ratio test, we compute \(\lim_{n \to \infty} |\frac{a_{n+1}} {a_n}|\). 根据比值判别法，我们计算 \(\lim_{n \to \infty} |\frac{a_{n+1}} {a_n}|\)。 |
| squeeze | /skwiːz/ | v./n. | the squeeze theorem 夹逼定理 | 夹在，夹逼 | The sequence is squeezed between two sequences that converge to the same limit. 该数列被夹在两个收敛到同一极限的数列之间。 |
Using the squeeze theorem, we show that the limit exists. 利用夹逼定理，我们证明该极限存在。 |
| bound | /baʊnd/ | n./v. | be bounded by 有界（上界或下界） | 界限 | Every convergent sequence is a bounded sequence. 每个收敛数列都是有界数列。 |
The sequence is bounded below by \(0\). 数列有下界 \(0\)。 |
| condition | /kənˈdɪʃən/ | n. | necessary condition 必要条件； sufficient condition 充分条件； convergence condition 收敛条件 |
条件 | The condition \(|r| \lt 1\) ensures the convergence of a geometric series. 条件 \(|r| \lt 1\) 保证几何级数收敛。 |
Under this condition, the series converges. 在该条件下级数收敛。 |
| notation | /nəʊˈteɪʃn/ | n. | 记号 | Notation helps simplify complex expressions. 符号表示简化复杂表达式。 |
We use standard notation for derivatives. 我们使用标准导数记号。 |
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| abbreviation | /əˌbriːviˈeɪʃn/ | n. | abbreviate v. | 缩写 | Abbreviation makes mathematical writing shorter. 缩写使数学书写更简洁。 |
“w.r.t.” is an abbreviation for with respect to. “w.r.t.”是with respect to的缩写。 |
| format | /ˈfɔːmæt/ | n. | 格式 | The format of equations must be consistent. 方程格式必须一致。 |
We follow a standard format for proofs. 我们遵循标准证明格式。 |
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| convention | /kənˈvenʃn/ | n. | 约定，惯例 | Convention determines standard notation. 约定决定标准符号。 |
We follow mathematical convention in signs. 我们遵循数学符号约定。 |
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| standard | /ˈstændəd/ | adj./n. | 标准 | This is the standard form of a quadratic. 这是二次函数的标准形式。 |
Standard methods are used in solving equations. 解方程使用标准方法。 |
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| customary | /ˈkʌstəməri/ | adj. | 惯例的 | Customary notation varies slightly between books. 不同书籍的惯用符号略有不同。 |
It is customary to write limits this way. 通常这样写极限。 |
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| ambiguity | /ˌæmbɪˈɡjuːəti/ | n. | 歧义 | We remove ambiguity by defining variables. 通过定义变量消除歧义。 |
Ambiguity arises from unclear notation. 不明确的符号会产生歧义。 |
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| distinction | /dɪˈstɪŋkʃn/ | n. | 区分 | The distinction is important in calculus. 这种区分在微积分中很重要。 |
We make a distinction between limit and value. 我们区分极限和值。 |
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| bracket | /ˈbrækɪt/ | n. | parenthesis 圆括号 | 括号 | Square brackets indicate intervals. 方括号表示区间。 |
We use brackets to group terms. 我们用括号分组。 |
