Taylor Series in A-Level Edexcel Mathematics: Key Points Explained | A-Level Edexcel 数学：泰勒级数考点精讲

📚 Taylor Series in A-Level Edexcel Mathematics: Key Points Explained | A-Level Edexcel 数学：泰勒级数考点精讲

Taylor series provide a systematic way to represent a wide range of functions as infinite power series. By extracting derivatives at a single point, we can build polynomial approximations that are invaluable for analysis, numerical estimation and solving differential equations. In the Edexcel A-Level Further Mathematics specification, a solid grasp of Maclaurin and Taylor expansions, standard series, and convergence criteria is essential for success.

泰勒级数提供了一种将多种函数表示为无穷幂级数的系统方法。通过提取函数在单点的各阶导数，我们可以构造多项式逼近，这对于分析、数值估算和求解微分方程都非常重要。在 Edexcel A-Level 进阶数学考试大纲中，牢固掌握麦克劳林展开、泰勒展开、标准级数以及收敛准则是取得好成绩的关键。

1. Introduction to Taylor Series | 泰勒级数简介

Taylor series express a differentiable function f(x) as an infinite sum of terms, each derived from the function’s derivatives at a chosen point a. They are a cornerstone of calculus because they allow complicated functions to be approximated by polynomials near that point, opening the door to easy computation and analysis.

泰勒级数将一个可微函数 f(x) 表示为无穷多项的和，每一项都来自于该函数在选定点 a 处的各阶导数。它们是微积分的基石，因为它们允许在这一点附近用多项式逼近复杂函数，从而使计算和分析变得简单。

2. Maclaurin Series | 麦克劳林级数

A Maclaurin series is the special case of a Taylor series centred at a = 0. The expansion is given by:

麦克劳林级数是泰勒级数在 a = 0 处的特殊情况。其展开式为：

f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + …

This series provides an exact representation of f(x) for all x inside the interval of convergence, provided f is infinitely differentiable at 0. To use it, you need to evaluate f and its successive derivatives at x = 0.

只要 f 在 0 处无限可导，且 x 在收敛区间内，这个级数就能精确地表示 f(x)。使用这一公式时，需要计算 f 及其各阶导数在 x = 0 处的值。

3. General Taylor Series | 一般泰勒级数

When we need to expand a function about a non‑zero point x = a, the general Taylor series is used:

当需要围绕非零点 x = a 展开函数时，就要使用一般的泰勒级数：

f(x) = f(a) + f'(a)(x-a) + f”(a)(x-a)²/2! + … + f⁽ⁿ⁾(a)(x-a)ⁿ/n! + …

This is particularly important for functions like ln x, which is undefined at 0, but can be expanded around a = 1. The choice of a often simplifies the derivatives or keeps the expansion inside a desired interval.

对于 ln x 等在 0 处无定义的函数，这一点尤为重要，但我们可以围绕 a = 1 进行展开。a 的选择通常能简化导数计算，或使展开保持在所需区间内。

4. Finding Derivatives for Expansions | 求导展开

To build a Taylor or Maclaurin series, you must compute derivatives systematically. Organise your work in a table that lists the function and its successive derivatives, then evaluate each at the expansion point. The table helps spot patterns and minimises algebraic errors.

要建立泰勒或麦克劳林级数，必须有条理地计算各阶导数。可以将函数及其各阶导数列成表格，然后在展开点逐一求值。表格有助于发现规律，并最大限度减少代数错误。

For composite functions, the chain rule and product rule are frequently needed. Practising the differentiation of eˣ sin x, ln(cos x) or similar functions sharpens the skills required in the exam.

对于复合函数，经常需要用到链式法则和乘法法则。多练习 eˣ sin x、ln(cos x) 等函数的求导，能有效提高考试所需的能力。

5. Standard Expansions | 标准展开式

Candidates are expected to know the following Maclaurin series by heart. They serve as building blocks for many exam questions.

考生需要熟记以下麦克劳林级数，它们是解答许多考题的基础。

The exponential function: eˣ = 1 + x + x²/2! + x³/3! + … + xⁿ/n! + …, valid for all real x.

指数函数：eˣ = 1 + x + x²/2! + x³/3! + … + xⁿ/n! + …，对所有实数 x 成立。

Sine: sin x = x − x³/3! + x⁵/5! − x⁷/7! + … + (−1)ⁿ x²ⁿ⁺¹/(2n+1)! + …, valid for all real x.

正弦：sin x = x − x³/3! + x⁵/5! − x⁷/7! + … + (−1)ⁿ x²ⁿ⁺¹/(2n+1)! + …，对所有实数 x 成立。

Cosine: cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … + (−1)ⁿ x²ⁿ/(2n)! + …, valid for all real x.

余弦：cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … + (−1)ⁿ x²ⁿ/(2n)! + …，对所有实数 x 成立。

Natural logarithm: ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … + (−1)ⁿ⁻¹ xⁿ/n + …, converging for −1 < x ≤ 1.

自然对数：ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … + (−1)ⁿ⁻¹ xⁿ/n + …，收敛区间为 −1 < x ≤ 1。

Binomial expansion: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …, valid for |x| < 1 when n is not a positive integer.

二项式展开：(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …，当 n 不是正整数时，适用范围为 |x| < 1。

6. Approximating Functions | 函数逼近

Truncating a Taylor series after a few terms gives a Taylor polynomial that approximates the function near the expansion point. For instance, near x = 0 we can use sin x ≈ x − x³/6. The accuracy improves as more terms are retained, especially when x is small.

在级数的若干项之后截断，就得到泰勒多项式，它能逼近展开点附近的函数。例如，在 x = 0 附近可用 sin x ≈ x − x³/6。当 x 较小时，保留的项数越多，精度就越高。

To estimate sin(0.1), compute 0.1 − (0.1)³/6 = 0.1 − 0.0001666… = 0.0998333…, which matches the true value to five decimal places. Such approximations are frequently tested.

估算 sin(0.1) 时，计算 0.1 − (0.1)³/6 = 0.1 − 0.0001666… = 0.0998333…，与真实值在小数点后五位内一致。这类近似估算在考试中经常出现。

7. Interval of Convergence | 收敛区间

A Taylor series does not necessarily converge for every x. The set of x for which the series converges to the function is called the interval of convergence. For A‑Level, the ratio test is the standard method to determine this interval.

泰勒级数并非对所有 x 都收敛。使级数收敛到原函数的所有 x 的集合称为收敛区间。在 A‑Level 中，比值判别法是确定收敛区间的标准方法。

For the Maclaurin series of ln(1+x), applying the ratio test shows convergence for −1 < x < 1 and divergence for |x| > 1. The endpoint x = 1 must be checked separately and yields conditional convergence.

对于 ln(1+x) 的麦克劳林级数，应用比值法可知在 −1 < x < 1 时收敛，|x| > 1 时发散。端点 x = 1 需要单独检验，结果是条件收敛。

Always state the interval of convergence when giving a series expansion. Some questions explicitly ask for the values of x for which the expansion is valid.

给出级数展开式时，务必说明收敛区间。有些题目会明确要求写出展开式有效的 x 取值范围。

8. Composite Functions | 复合函数展开

It is often possible to obtain a series for a composite function by substituting one known expansion into another. This avoids tedious differentiation. For example, to expand eˢⁱⁿ ˣ near 0, substitute the series for sin x into the series for eˣ:

很多时候，可以将一个已知展开式代入另一个已知展开式而得到复合函数的级数，从而避免繁琐的求导工作。例如，在 0 附近展开 eˢⁱⁿ ˣ，可将 sin x 的级数代入 eˣ 的级数：

Let u = x − x³/6 + …, then eᵘ = 1 + u + u²/2! + … . Expanding and keeping terms up to x³ gives eˢⁱⁿ ˣ = 1 + x + x²/2 − x⁴/8 + … (terms re‑collected). This technique saves time and is a favourite in exams.

令 u = x − x³/6 + …，则 eᵘ = 1 + u + u²/2! + … 。展开并保留到 x³ 项可得 eˢⁱⁿ ˣ = 1 + x + x²/2 − x⁴/8 + … （合并各项）。这种方法节省时间，是考试中的高频技巧。

Substitution must be valid within the overlap of convergence intervals of both series. Always check that the final expansion is correct within the desired range.

代入必须在两个级数收敛区间的交集内有效。务必验证最终展开式在所需范围内正确。

9. Error Term and Lagrange Remainder | 误差项与拉格朗日余项

When a Taylor series is truncated after the term containing (x−a)ⁿ, the error introduced can be estimated by the Lagrange remainder:

当泰勒级数在含有 (x−a)ⁿ 的项之后截断时，引入的误差可用拉格朗日余项来估计：

Rₙ = f⁽ⁿ⁺¹⁾(c) (x−a)ⁿ⁺¹/(n+1)!

where c lies somewhere between a and x. This remainder provides an upper bound on the absolute error, allowing us to determine how many terms are needed for a desired accuracy.

其中 c 介于 a 与 x 之间。这个余项给出了绝对误差的上界，使我们能够确定达到指定精度所需要的项数。

In an exam, you might be asked to show that the error in using a third‑degree Taylor polynomial for a certain function is less than a given value, by finding the maximum of |f⁽⁴⁾(x)| on the interval. Keep in mind the conditions for the remainder theorem.

在考试中，可能会要求你证明使用某函数的三次泰勒多项式时，误差小于给定的数值，这需要找出区间上 |f⁽⁴⁾(x)| 的最大值。请牢记余项定理的适用条件。

10. Applications and Exam Tips | 应用与考试技巧

Typical examination tasks include: deriving the Maclaurin series up to a specified term, evaluating limits using series expansions (for example, lim(x→0) (sin x − x)/x³), and approximating integrals by replacing the integrand with its Taylor polynomial. Practice these applications repeatedly.

典型的考试任务包括：推导指定阶数的麦克劳林级数，利用级数展开式求极限（例如 lim(x→0) (sin x − x)/x³），以及将被积函数替换为其泰勒多项式以近似计算积分。应反复练习这些类型的题目。

Always present your working clearly: show the table of derivatives, write the general term if possible, and state the series with factorial denominators. Simplify coefficients and explicitly note the interval of convergence. Marks are awarded for method and clarity.

解答过程务必清晰：列出导数表格，如有可能写出通项，并用阶乘作为分母列写级数。简化系数，并明确注出收敛区间。评分会考虑书写方法与条理性。

Finally, check your answer by substituting a small value and comparing it with a calculator result. This can highlight algebraic slips and boost confidence.

最后，取一个小值代入您的展开式，与计算器结果进行比较，可以检验是否有代数错误，并增强信心。

Published by TutorHao | Further Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Taylor Series in A-Level Edexcel Mathematics: Key Points Explained | A-Level Edexcel 数学：泰勒级数 考点精讲