An Introduction to Series | 级数导论

📚 An Introduction to Series | 级数导论

A series is one of the most practical topics in Cambridge Mathematics because it connects patterns in numbers with powerful summation formulas. This introduction covers the key ideas you need before working with arithmetic and geometric progressions, sigma notation, and sums to infinity.

级数是剑桥数学中最实用的主题之一，因为它将数字规律与强大的求和公式联系起来。本导论涵盖在学习等差数列、等比数列、求和符号以及无穷和之前需要掌握的关键概念。

1. The Building Blocks: Sequences and Series Terminology | 基本概念：数列与级数的术语

A sequence is an ordered list of numbers written as u₁, u₂, u₃, …, uₙ, where each number is called a term. The position of a term is shown by the subscript, so u₁ is the first term and uₙ is the n-th term.

数列是按一定顺序排列的数，记作 u₁, u₂, u₃, …, uₙ，其中每个数称为项。项的位置由下标表示，因此 u₁ 是首项，uₙ 是第 n 项。

A series is what you get when you add the terms of a sequence. The expression u₁ + u₂ + u₃ + … + uₙ represents a finite series with n terms. If the addition continues forever, it is an infinite series.

级数是将数列的项相加所得的结果。表达式 u₁ + u₂ + u₃ + … + uₙ 表示含有 n 项的有限级数。如果相加一直持续，则为无限级数。

In Cambridge Mathematics, the word series is often used even when the main focus is the sum formula. You need to move confidently between the individual terms of a sequence and the series formed from them.

在剑桥数学中，即使重点在求和公式，也经常使用“级数”一词。你需要能够在数列的项和由它们形成的级数之间自如转换。

2. Finite and Infinite Series | 有限级数与无限级数

A finite series has a definite number of terms. For example, the series 1 + 2 + 3 + … + 100 has exactly 100 terms, and its sum can be calculated directly or by using a formula.

有限级数有确定的项数。例如级数 1 + 2 + 3 + … + 100 恰好有 100 项，其和可以直接计算或用公式求出。

An infinite series continues without end. Some infinite series, such as 1 + 1/2 + 1/4 + 1/8 + …, get closer and closer to a fixed value, while others, such as 1 + 2 + 3 + …, grow without bound.

无限级数没有终止。有些无限级数，如 1 + 1/2 + 1/4 + 1/8 + …，会越来越接近一个固定值；有些，如 1 + 2 + 3 + …，则会无限增大。

In A-level problems, finite series are usually arithmetic or geometric. Infinite series appear mostly as geometric series with a sum to infinity.

在 A-level 题目中，有限级数通常为等差或等比级数。无限级数主要作为具有无穷和的等比级数出现。

3. Sigma Notation | 求和符号 Σ

Sigma notation is a shorthand way to write series. The Greek letter Σ means sum. The general term is written after Σ, and the lower and upper limits show where the sum starts and ends.

求和符号 Σ 是级数的简写。希腊字母 Σ 表示求和。通项写在 Σ 之后，下限和上限表示求和的起止位置。

For instance, Σₖ₌₁⁵ k² means 1² + 2² + 3² + 4² + 5². Here k is the index of summation, 1 is the lower limit, and 5 is the upper limit.

例如 Σₖ₌₁⁵ k² 表示 1² + 2² + 3² + 4² + 5²。这里 k 是求和变量，1 是下限，5 是上限。

Σₖ₌₁ⁿ aₖ = a₁ + a₂ + a