📚 A note on infinite series | 无穷级数笔记
Infinite series are fundamental objects in higher mathematics, forming the basis for calculus, differential equations, and much of modern analysis. For IB Mathematics students, mastering infinite series means understanding convergence, using tests, and applying series such as geometric, p‑series, Taylor and Maclaurin expansions. This note consolidates the essential theory, practical tests, and examples needed for examinations.
无穷级数是高等数学的基础对象，构成了微积分、微分方程和现代分析的重点。对于IB数学学生而言，掌握无穷级数意味着理解收敛性，运用各种判别法，以及应用几何级数、p级数、泰勒级数和麦克劳林展开式。本文整理了考试所需的基本理论、实用检验方法和例题。
1. Sequences and Series: Definitions | 数列与级数：定义
An infinite sequence is an ordered list of numbers a₁, a₂, a₃, ... , aₙ, ... continuing without end. An infinite series is the sum of all terms of an infinite sequence, denoted by Σn=1∞ an or simply Σ an. Because we cannot add infinitely many numbers by ordinary arithmetic, we approach series via partial sums. The Nth partial sum SN is defined as SN = Σn=1N an = a₁ + a₂ + ... + aN. If the sequence of partial sums {SN} converges to a finite limit L, we say the series converges to L, written Σ an = L. Otherwise it diverges.
无穷数列是一个没有尽头的有序数表 a₁, a₂, a₃, ... , aₙ, ... 。无穷级数就是这个数列所有项的和，记作 Σn=1∞ an 或简写为 Σ an。由于我们无法用普通的加法直接相加无限多个数，因此我们通过部分和来处理级数。第 N 部分和 SN 定义为 SN = Σn=1N an = a₁ + a₂ + ... + aN。如果部分和数列 {SN} 收敛到一个有限的极限 L，我们就说该级数收敛于 L，记作 Σ an = L。否则级数发散。
2. Partial Sums and Convergence | 部分和与收敛
Convergence of an infinite series is determined by the behaviour of its partial sums SN as N → ∞. If limN→∞ SN exists and equals a real number S, the series converges to S. If the limit does not exist (for example, it oscillates) or is infinite, the series diverges. It is crucial to distinguish between the sequence of terms {an} and the sequence of partial sums {SN}. Even if the terms an tend to zero, the series may still diverge, as with the harmonic series.
无穷级数的收敛性由它的部分和 SN 在 N → ∞ 时的行为决定。如果 limN→∞ SN 存在且等于一个实数 S，该级数收敛于 S。如果极限不存在（例如振荡）或是无穷大，级数发散。区分项数列 {an} 和部分和数列 {SN} 至关重要。即使通项 an 趋于零，级数仍可能发散，调和级数就是一个例子。
3. Geometric Series | 几何级数
A geometric series has the form Σn=0∞ arn + ar³ + ... , where a is the first term and r is the common ratio. The series converges if |r| < 1, in which case its sum is a/(1 − r). If |r| f the series diverges. This is one of the few series for which an exact formula for the sum exists. IB exam questions frequently require students to identify the common ratio, check convergence, and compute the sum.
几何级数的形式为 Σn=0∞ arn = a + ar + ar² + ar³ + ... ，其中 a 是首项，r 是公比。当 |r| < 1 时级数收敛，其和为 a/(1 − r)。若 |r| ≥ 1，级数发散。这是为数不多的有精确求和公式的级数之一。IB 试题常要求学生识别公比，检验收敛性并计算总和。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图