📚 AS Mathematics: Taylor Series Essentials | AS数学：泰勒级数考点精讲
Taylor series provide an elegant way to represent a wide class of functions as infinite polynomials. In AS Mathematics, mastering the fundamentals of Taylor and Maclaurin expansions is essential for function approximation, limit evaluation, and understanding local behaviour of functions. This article covers key concepts, worked examples and exam-style tips to build confidence with series expansions.
泰勒级数提供了一种将众多函数表示为无穷多项式的优雅方法。在AS数学中，掌握泰勒与麦克劳林展开的基础知识，对于函数逼近、极限计算以及理解函数的局部行为至关重要。本文涵盖核心概念、典型例题与考试技巧，助你扎实掌握级数展开。
1. What is a Taylor Series? | 什么是泰勒级数？
The Taylor series of a smooth function f(x) about a point x = a is an infinite sum built from the values of the function and its derivatives at a. The idea is to construct a polynomial that matches f and all its derivatives at the chosen centre, providing excellent local approximations.
光滑函数 f(x) 在点 x = a 处的泰勒级数是一个由该点函数值及其各阶导数构成的无穷级数。其核心思想是构造一个多项式，使它在所选中心处与原函数及其所有阶导数完全吻合，从而给出极好的局部逼近。
f(x) = f(a) + f'(a)(x−a) + f”(a)/2! (x−a)² + f”'(a)/3! (x−a)³ + ⋯ + f⁽ⁿ⁾(a)/n! (x−a)ⁿ + ⋯
此公式中的每一项 (x−a)ᵏ 乘以对应的 k 阶导数值并除以 k!。若取前有限项，我们便得到泰勒多项式，用于实际计算中的近似。
2. Maclaurin Series: Expansion at Zero | 麦克劳林级数：在零点展开
When the centre is chosen as a = 0, the Taylor series simplifies and is called a Maclaurin series. This special case is the most common in AS examinations and forms the basis for standard expansions of elementary functions.
当展开中心取为 a = 0 时，泰勒级数得到简化，称为麦克劳林级数。这一特例是AS考试中最常见的情形，并构成了基本初等函数标准展开式的基础。
f(x) = f(0) + f'(0)x + f”(0)/2! x² + f”'(0)/3! x³ + ⋯
麦克劳林级数将函数在原点附近展开成 x 的幂级数。对于多数标准函数，只要原函数在零处无限可导，我们都能写出其无穷展开。
3. Maclaurin Series for eˣ | eˣ 的麦克劳林级数
The exponential function f(x)=eˣ is the simplest non-polynomial function to expand because all its derivatives are eˣ, and at x=0 they all equal 1.
指数函数 f(x)=eˣ 是最容易展开的非多项式函数，因为它的所有阶导数都是 eˣ，且在 x=0 处它们都等于 1。
eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ⋯ = ∑ xⁿ/n! (n=0 to ∞)
该级数对全体实数 x 收敛——即收敛半径无穷大。因此我们可以在任意精度下用多项式逼近 eˣ。
4. Maclaurin Series for sin x and cos x | sin x 与 cos x 的麦克劳林级数
The sine and cosine functions yield alternating series whose terms involve only odd or even powers of x, respectively. Their derivatives cycle every four steps, making the pattern easy to remember.
正弦和余弦函数给出的都是交错级数，且分别只含 x 的奇次幂或偶次幂。它们的导数每四步循环一次，规律易于记忆。
sin x = x − x³/3! + x⁵/5! − x⁷/7! + ⋯
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ⋯
两个级数对全体实数都收敛。注意到 sin x 的展开保留了奇函数性质，而 cos x 保留了偶函数性质，这与函数本身对称性一致。
5. Maclaurin Series for ln(1+x) | ln(1+x) 的麦克劳林级数
The natural logarithm shifted by 1 has derivatives that produce alternating signs and factorials in denominators. The expansion is valid only for a limited interval.
平移后的自然对数函数 ln(1+x) 其导数生成交错符号与分母阶乘的组合。此展开仅在一个有限区间内有效。
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + ⋯ + (−1)ⁿ⁻¹ xⁿ/n + ⋯
收敛区间为 −1 < x ≤ 1。特别注意在端点 x=1 处级数收敛到 ln 2，而在 x=−1 处发散。
6. Maclaurin Series for (1+x)ⁿ | (1+x)ⁿ 的二项式展开
For any real exponent n, the binomial series generalises the ordinary binomial theorem. The coefficients involve falling factorials and factorial denominators.
对于任意实数指数 n，二项式级数推广了普通二项式定理，其系数包含下降阶乘与阶乘分母。
(1+x)ⁿ = 1 + n x + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ⋯
当 n 不是正整数时，级数收敛于 |x| < 1。例如 √(1+x) = 1 + ½ x − 1/8 x² + ⋯ 是常用的有理指数情形。
7. Taylor Polynomials and Order of Approximation | 泰勒多项式与逼近阶数
Truncating a Taylor series after a finite number of terms gives a Taylor polynomial. The degree of the polynomial determines how well it approximates the original function near the centre.
将泰勒级数在有限项后截断就得到一个泰勒多项式。多项式的次数决定了它在中心附近逼近原函数的精度。
Pₙ(x) = f(a) + f'(a)(x−a) + ⋯ + f⁽ⁿ⁾(a)/n! (x−a)ⁿ
例如，用二次泰勒多项式逼近 eˣ 在 x=0 附近：P₂(x)=1+x+x²/2。增加次数可快速减小误差，尤其在 x 较小时。
8. The Lagrange Remainder and Error Bound | 拉格朗日余项与误差界
The error when approximating a function by its Taylor polynomial can be expressed by the Lagrange remainder. This formula provides an upper bound for the absolute error, which is central to justifying approximations.
用泰勒多项式逼近函数时的误差可由拉格朗日余项表示。该公式给出了绝对误差的上界，是论证近似合理性的关键依据。
Rₙ(x) = f⁽ⁿ⁺¹⁾(c) / (n+1)! · (x−a)ⁿ⁺¹
其中 c 是介于 a 与 x 之间的某个数。为确定最大可能误差，通常取 |f⁽ⁿ⁺¹⁾(c)| 在区间上的最大值代入。
9. Determining the Radius and Interval of Convergence | 确定收敛半径与收敛区间
Not all Maclaurin series converge for all x. The radius of convergence R tells us the range |x| 并非所有麦克劳林级数都对全体 x 收敛。收敛半径 R 给出了使级数收敛的范围 |x| For series ∑ aₙ xⁿ, R = 1 / lim sup |aₙ|¹ᐟⁿ 常见结果：eˣ, sin x, cos x 的收敛半径 R=∞；ln(1+x) 和 (1+x)ⁿ 的收敛半径 R=1。端点需要单独检验。 Taylor expansions are powerful tools for evaluating indeterminate limits of the form 0/0. By expanding the numerator and denominator around the limit point, the leading-order terms often cancel, revealing the finite limit. 泰勒展开是计算 0/0 型未定式极限的强大工具。将分子和分母在极限点附近展开，高阶项通常会抵消，从而显现出有限极限值。 Example: lim_{x→0} (eˣ − 1 − x) / x². Substituting the series eˣ = 1 + x + x²/2 + x³/6 + ⋯ gives numerator x²/2 + x³/6 + ⋯, so the limit is 1/2. 例如：lim_{x→0} (eˣ − 1 − x) / x²。代入展开 eˣ = 1 + x + x²/2 + x³/6 + ⋯，分子变为 x²/2 + x³/6 + ⋯，从而极限为 1/2。 Use a second-degree Taylor polynomial for f(x)=√x about a= Published by TutorHao | Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
10. Using Taylor Series to Evaluate Limits | 使用泰勒级数求极限
11. Worked Example | 实例解析
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