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Taylor Series Revision | 泰勒级数考点精讲

2026-07-13 21:35 · 本文部分内容由AI辅助创作

📚 Taylor Series Revision | 泰勒级数考点精讲

In CCEA GCSE Further Mathematics, Taylor series provides a powerful way to represent functions as infinite polynomials. At this level, you are expected to understand and use the Maclaurin series (Taylor series centred at 0) for the exponential, sine, and cosine functions. This article covers the key expansions, how to approximate values, and common exam techniques.

在 CCEA GCSE 进阶数学中，泰勒级数是一种将函数表示为无限多项式的重要方法。在这个级别，你需要理解并应用指数函数、正弦函数和余弦函数的麦克劳林级数（即以 0 为中心的泰勒级数）。本文涵盖关键的展开式、如何近似计算以及常见的考试技巧。

1. What is a Taylor Series? | 什么是泰勒级数？

tives at a single point. If the point is x = 0, the expansion is called a Maclaurin series – and that is the only form you will meet in GCSE Further Maths.

泰勒级数将函数 f(x) 展开为无穷多项的和，这些项由函数在某个单点处的各阶导数值决定。如果这个点是 x = 0，那么展开式就称为麦克劳林级数——这也是你在 GCSE 进阶数学中唯一会见到的形式。

At this level, you are not required to derive the series using differentiation. Instead, you memorise the standard expansions and use them to approximate function values or to identify the function from a given series.

在这个级别，你不需要用微分来推导级数。你只需记住标准展开式，并用它们来近似函数值，或者根据给出的级数识别出原函数。

2. Maclaurin Series – Taylor at Zero | 麦克劳林级数——在零点的泰勒级数

The general form of a Maclaurin series is f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! er Maths, three specific expansions are essential. They are summarised in the table below.

麦克劳林级数的一般形式为 f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... 。在 CCEA GCSE 进阶数学中，有三个特定的展开式至关重要。下表对它们进行了总结。

Function Maclaurin Series (first non‑zero terms)
eˣ 1 + x + x²/2! + x³/3! + x⁴/4! + ...
sin x x − x³/3! + x⁵/5! − x⁷/7! + ...
cos x 1 − x²/2! + x⁴/4! − x⁶/6! + ...

Notice the alternating signs in the sine and cosine series, while the exponential series has all positive terms. Also, x must be in radians when using the series for sin x and cos x.

注意正弦和余弦级数的符号是交替的，而指数级数的各项均为正。此外，当使用 sin x 和 cos x 的级数时，x 必须以弧度为单位。

3. The Maclaurin Series for eˣ | eˣ 的麦克劳林展开式

The expansion for eˣ is the simplest because every derivative at zero equals 1. The series is:

eˣ 的展开式是最简单的，因为其在零点的所有导数都等于 1。级数为：

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ... + xⁿ/n! + ...

Each term is the power of x divided by its factorial. This series converges for all real values of x, so you can use it to approximate e raised to any power.

每一项是 x 的幂除以其阶乘。这个级数对所有实数 x 都收敛，因此你可以用它来近似 e 的任意次幂。

4. The Maclaurin Series for sin x | sin x 的展开式

The sine series contains only odd powers of x, and the signs alternate, starting with a positive sign. The expansion is:

正弦级数只包含 x 的奇次幂，符号交替出现，且以正号开始。展开式为：

sin x = x − x³/3! + x⁵/5! − x⁷/7! + ...

Because the series involves factorials, the denominators grow very quickly. Even a few terms can give a very accurate approximation when x is small.

由于级数中包含阶乘，分母增长非常快。当 x 较小时，只需少数几项就能给出非常精确的近似值。

5. The Maclaurin Series for cos x | cos x 的展开式

In contrast, the cosine series has only even powers of x, and the signs also alternate, beginning with a positive sign. The standard form is:

相反地，余弦级数只包含 x 的偶次幂，符号同样交替，并以正号开始。标准形式为：

cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ...

You can think of cos x as an 'even function', which is why its series lacks odd powers. Both sin x and cos x expansions are valid for all x, though practical approximations are best for |x| small.

你可以将 cos x 视为“偶函数”，这就是它的级数没有奇次项的原因。sin x 和 cos x 的展开式对所有 x 都有效，但当 |x| 较小时近似效果最佳。

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6. Using Series to Approximate Values | 使用级数近似求值

To approximate a function value, substitute the given x into the expansion and take the sum of the first few terms. The more terms you include, the closer the sum gets to the true value. In exam questions, you will often be told how many terms to use, for example, 'up to and including the term in x³'.

要近似某个函数值，将给定的 x 代入展开式，并取前几项的和。包含的项数越多，和就越接近真实值。在考试题中，通常会明确要求你使用的项数，例如“直到并包含 x³ 项”。

Always keep x in radians for trigonometric series. Convert degrees to radians by multiplying by π/180 if necessary, though most CCEA questions will give x in radians directly.

使用三角级数时务必保持 x 以弧度为单位。如有必要，可将度数乘以 π/180 转换为弧度，不过大多数 CCEA 题目会直接给出弧度值。

7. Worked Example: Approximating e | 例题：近似计算自然常数 e

To estimate the value of e (which is e¹), set x = 1 in the eˣ series. Using terms up to 1/5! gives:

为了估算 e 的值（即 e¹），将 x = 1 代入 eˣ 的级数。使用直到 1/5! 的项：

e ≈ 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! = 1 + 1 + 0.5 + 0.166666... + 0.041666... + 0.008333...

Adding these gives 2.71666... (to 5 decimal places). The true value of e is approximately 2.71828, so the error is only about 0.0016. Adding the 1/6! term would improve it further.

将这些相加得到 2.71666...（精确到小数点后五位）。e 的真实值约为 2.71828，因此误差仅约 0.0016。若再加上 1/6! 项，精度会进一步提高。

8. Approximating sin and cos | 正弦和余弦的近似计算

Let' ate (0.2)³ = 0.008, divide by 6 = 0.001333..., so sin(0.2) ≈ 0.2 − 0.001333... = 0.198667.

我们来近似计算 sin(0.2)。使用直到 x³ 项的级数：sin(0.2) ≈ 0.2 − (0.2)³/6。计算 (0.2)³ = 0.008，除以 6 得 0.001333...，因此 sin(0.2) ≈ 0.2 − 0.001333... = 0.198667。

0.955 gives a quick estimate; the true value is 0.955336..., showing impressive accuracy.

sin(0.2) 的真实值约为 0.198669，可见即使只取两项，近似已经非常出色。对于 cos(0.3)，使用 1 − (0.3)²/2 = 1 − 0.045 = 0.955 即可快速估算；真实值为 0.955336...，精度令人印象深刻。

9. Number of Terms and Accuracy | 项数与精度

The accuracy of a Taylor approximation depends on how many terms you use and the size of x. For small x (e.g. 0.1, 0.2), the high powers of x become very small, so the series converges rapidly. Even the first two or three non‑zero terms can yield answers correct to several decimal places.

泰勒近似的精度取决于使用的项数以及 x 的大小。对于较小的 x（如 0.1、0.2），x 的高次幂变得非常小，因此级数收敛很快。即使只取前两到三个非零项，也可以得到精确到小数点后几位的答案。

In an exam, you might be asked to approximate a value to a certain degree of accuracy, or to state whether a given partial sum overestimates or underestimates the true value. For alternating series like sin x and cos x, the error after stopping at a term is less than the absolute value of the next term.

考试中，可能会要求你将数值近似到指定精度，或判断某个部分和是高估还是低估了真实值。对于像 sin x 和 cos x 这样的交错级数，在某一项停止后的误差小于下一项的绝对值。

10. Recognising Expansions | 识别展开式

Another common question type gives a series and asks you to identify the function it represents. For example, 1 − x²/2! + x⁴/4! − x⁶/6! + ... is clearly the Maclaurin series for cos x. A series like x + x³/3! + x⁵/5! + ... might look similar but lacks alternating signs – that would not be a standard GCSE series; the presence of all positive signs suggests an expression like (eˣ − e⁻ˣ)/2, but you are not expected to know that. Stick to the three basic patterns.

另一种常见题型是给出一个级数，要求你识别它所代表的函数。例如，1 − x²/2! + x⁴/4! − x⁶/6! + ... 显然是 cos x 的麦克劳林级数。像 x + x³/3! + x⁵/5! + ... 这样的级数看似相似但没有正负交替——它不属于 GCSE 标准级数；所有项为正暗示了像 (eˣ − e⁻ˣ)/2 这样的表达式，但你不必掌握。只需紧扣三个基本模式。

Always check the pattern of signs, whether powers are even or odd, and the factorial denominators. A quick comparison with the table in Section 2 will identify the correct function.

始终检查符号模式、幂次是奇是偶以及阶乘分母。与第 2 节表格快速对比就能确定正确的函数。

11. Exam Tips for CCEA GCSE Further Maths | CCEA GCSE 进阶数学考试技巧