Series and Limits | 级数与极限

📚 Series and Limits | 级数与极限

In AQA A-level Mathematics, Series and Limits covers arithmetic and geometric sequences, summation techniques, recurrence relations, binomial series and limiting behaviour. The topic tests both routine formula use and the ability to justify convergence in infinite cases.

在 AQA A-level 数学中，“级数与极限”涵盖等差与等比数列、求和技巧、递推关系、二项级数以及极限行为。本专题既考查公式的常规使用，也考查在无穷情形下证明收敛性的能力。

1. Sequences and Notation | 数列与记号

A sequence is an ordered list of numbers u₁, u₂, u₃, … produced by a rule. The nth term is written uₙ or aₙ, and the first term is often denoted u₁ or a.

数列是按规则产生的一列有序数字 u₁、u₂、u₃ ……。第 n 项写作 uₙ 或 aₙ，首项通常记为 u₁ 或 a。

A series is the sum of the terms of a sequence. The sum of the first n terms is written Sₙ = u₁ + u₂ + … + uₙ. The symbol ∑ is used as a compact summation notation.

级数是数列各项之和。前 n 项和写作 Sₙ = u₁ + u₂ + …… + uₙ。符号 ∑ 用于紧凑地表示求和。

For example, if uₙ = 2n + 1, the sequence begins 3, 5, 7, 9, … and S₄ = 3 + 5 + 7 + 9 = 24.

例如，若 uₙ = 2n + 1，则数列开头为 3、5、7、9 ……，且 S₄ = 3 + 5 + 7 + 9 = 24。

2. Arithmetic Series | 等差级数

An arithmetic sequence has a constant difference d between consecutive terms. If the first term is a, then the nth term is uₙ = a + (n − 1)d.

等差数列的相邻项之差 d 为常数。若首项为 a，则第 n 项为 uₙ = a + (n − 1)d。

The sum of the first n terms is given by:

前 n 项和公式为：

Sₙ = n/2 × (a + L) = n/2 × [2a + (n − 1)d]

where L = a + (n − 1)d is the last term. The first form is useful when the last term is known.

其中 L = a + (n − 1)d 是末项。当已知末项时，第一种形式更实用。

The formula comes from pairing terms from the two ends: (a + L), (a + d + L − d), and so on. There are n/2 such equal pairs.

该公式来自将首尾项配对：(a + L)、(a + d + L − d) 等。这样的等和配对共有 n/2 对。

Example: Find the sum of the first 50 terms of 3, 7, 11, … Here a = 3, d = 4, n = 50, so S₅₀ = 50/2 [2(3) +

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）