📚 The Basic Series Expansions | 基本级数展开
In this article, we explore the key series expansions required for AQA A-Level Mathematics. These expansions allow us to represent functions such as exponentials, logarithms, and trigonometric functions as infinite polynomials, enabling approximation and analysis.
在本文中，我们探讨AQA A-Level数学所需的关键级数展开。这些展开使我们能够将指数函数、对数函数和三角函数表示为无穷多项式，从而进行近似与分析。
1. Introduction to Series Expansions | 级数展开概述
A series expansion expresses a function as an infinite sum of terms with increasing powers of a variable. For many functions, this provides a powerful way to compute values and understand behaviour near a chosen point.
级数展开将一个函数表示为按变量幂次递增的无穷项之和。对于许多函数，这提供了一种计算数值并理解在选定点附近行为的强大方法。
The most common types at A-Level are the binomial expansion, the exponential series, and the trigonometric and logarithmic series derived from Maclaurin’s theorem.
A-Level中最常见的类型是二项式展开、指数级数，以及由麦克劳林定理导出的三角函数和对数函数级数。
f(x) = f(0) + x f ‘(0) + x²⁄2! f ”(0) + x³⁄3! f ”'(0) + …
2. Binomial Expansion for Positive Integer n | 正整数 n 的二项式展开
For a positive integer n, the binomial theorem states that (a+b)ⁿ can be expanded as a sum of terms involving combinations. The coefficients are given by Pascal’s triangle or the notation ⁿCᵣ.
对于正整数 n，二项式定理指出 (a+b)ⁿ 可以展开为涉及组合数的各项之和。系数由帕斯卡三角或记号 ⁿCᵣ 给出。
(a+b)n = ∑ r=0n ⁿCr an−r br
When a = 1 and b = x, the expansion simplifies to (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + ⋯ + xⁿ. This form is essential for approximations when x is small.
当 a = 1 且 b = x 时，展开式简化为 (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + … + xⁿ。当 x 很小时，这种形式对于近似计算至关重要。
3. General Binomial Series | 一般二项式级数
For any real value of n, provided |x| < 1, the expression (1+x)ⁿ can be expanded as an infinite series. This is called the general binomial series.
对于任意实数 n，只要 |x| < 1，表达式 (1+x)ⁿ 就可以展开为无穷级数。这称为一般二项式级数。
(1+x)n = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …
Note that when n is a positive integer, the series terminates after n+1 terms. When n is not a positive integer, the series is infinite and converges only for |x| < 1.
注意，当 n 为正整数时，级数在 n+1 项后终止。当 n 不是正整数时，级数为无穷级数，且仅在 |x| < 1 时收敛。
This expansion is often used to find approximations to roots and reciprocals, such as 1/√(1+x) or 1/(1+x)².
这种展开常用于求根和倒数的近似值，例如 1/√(1+x) 或 1/(1+x)²。
4. Maclaurin Series for eˣ | 指数函数 eˣ 的麦克劳林级数
The exponential function eˣ has a Maclaurin series that converges for all real x. It is given by an infinite sum of powers divided by factorials.
指数函数 eˣ 的麦克劳林级数对所有实数 x 收敛。它由幂次除以阶乘的无穷和给出。
ex = 1 + x + x²/2! + x³/3! + ⋯ + xⁿ/n! + ⋯
This series is extremely useful for evaluating eˣ near x = 0, and also forms the basis for hyperbolic function expansions.
这个级数对于计算 x 接近 0 时的 eˣ 非常有用，也是双曲函数展开的基础。
5. Maclaurin Series for ln(1+x) | 对数函数 ln(1+x) 的麦克劳林级数
The natural logarithm ln(1+x) can be expanded as a series valid for −1 < x ≤ 1. The series is obtained by term-by-term integration of a geometric series.
自然对数 ln(1+x) 可以展开为在 −1 < x ≤ 1 内有效的级数。该级数是对几何级数逐项积分得到的。
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … for −1 < x ≤ 1
Note that the series does not converge at x = −1, but does converge at x = 1, giving ln 2 = 1 − 1/2 + 1/3 − 1/4 + …
注意该级数在 x = −1 处不收敛，但在 x = 1 处收敛，得到 ln 2 = 1 − 1/2 + 1/3 − 1/4 + …
6. Maclaurin Series for sin x and cos x | 正弦和余弦函数的麦克劳林级数
For trigonometric functions, the Maclaurin series involve only odd or even powers. The series for sin x and cos x converge for all real x.
对于三角函数，麦克劳林级数只涉及奇次或偶次幂。sin x 和 cos x 的级数对所有实数 x 收敛。
sin x = x − x³/3! + x⁵/5! − x⁷/7! + …
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …
These series are useful for small-angle approximations, and they reveal the relationship between exponential and trigonometric functions through Euler’s formula.
这些级数对小角度近似很有用，并且通过欧拉公式揭示了指数函数与三角函数之间的联系。
7. Using Series to Find Limits | 用级数求极限
Series expansions can be used to evaluate limits that are otherwise indeterminate. By expanding functions around a critical point, we can compare leading terms.
级数展开可用于计算原本不确定的极限。通过在临界点附近展开函数，我们可以比较首项。
For example, the limit of (sin x)/x as x → 0 is 1, which can be confirmed by dividing the series for sin x by x.
例如，当 x → 0 时，(sin x)/x 的极限为 1，这可以通过将 sin x 的级数除以 x 来确认。
sin x / x = 1 − x²/3! + x⁴/5! − … → 1 as x → 0
8. Combining Standard Series | 组合标准级数
Many functions can be expanded by substituting into known series. For example, e⁻ˣ² is obtained by replacing x with −x² in the exponential series.
许多函数可以通过代入已知级数来展开。例如，e⁻ˣ² 是将指数级数中的 x 替换为 −x² 得到的。
Similarly, ln(1+2x) uses the logarithmic series with 2x, valid for |2x| < 1, i.e. −1/2 < x < 1/2.
类似地，ln(1+2x) 使用带有 2x 的对数级数，其有效范围为 |2x| < 1，即 −1/2 < x < 1/2。
This technique is efficient and avoids differentiating complex functions repeatedly.
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导