Taylor Series | 泰勒级数考点精讲

📚 Taylor Series | 泰勒级数考点精讲

Taylor series provide a powerful way to represent many functions as infinite sums of polynomial terms. They allow us to approximate complicated curves with simple, manageable polynomials near a given point, which is essential in both pure mathematics and real-world applications like physics and engineering. Mastering Taylor series opens the door to understanding how calculators compute values of trigonometric, exponential and logarithmic functions.

泰勒级数提供了一种将许多函数表示为无穷多项式之和的强大方法。它们使我们能够在给定点附近用简单、易于处理的多项式来逼近复杂的曲线，这在纯数学以及物理、工程等实际应用中至关重要。掌握泰勒级数可以帮你理解计算器如何计算三角函数、指数函数和对数函数的值。

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

A Taylor series expands a function f(x) around a point x = a. If we know all the derivatives of f at a, we can write f(x) as an infinite power series. The core idea is to match the function’s value, slope, curvature and higher-order rates of change at that single point using polynomial terms.

泰勒级数围绕点 x = a 将函数 f(x) 展开。如果我们知道 f 在 a 处的所有导数，就可以将 f(x) 写成一个无穷幂级数。核心思想是利用多项式项在单个点上匹配函数的值、斜率、曲率以及更高阶的变化率。

2. Maclaurin Series as a Special Case | 麦克劳林级数：特殊情形

When the expansion point is a = 0, the Taylor series is called a Maclaurin series. This is by far the most common form encountered in IGCSE and beyond because many standard functions have neat Maclaurin expansions. For example, the Maclaurin series for eˣ, sin x and cos x all look elegant and symmetric.

当展开点 a = 0 时，泰勒级数被称为麦克劳林级数。这是目前 IGCSE 及更高阶段最常遇到的形式，因为许多标准函数都具有简洁的麦克劳林展开式。例如，eˣ、sin x 和 cos x 的麦克劳林级数都显得优美且对称。

3. Formula and Notation | 公式与符号

The general Taylor series of f(x) about x = a is:

函数 f(x) 关于 x = a 的一般泰勒级数为：

f(x) = f(a) + f'(a)(x – a) + f”(a)(x – a)² / 2! + f”'(a)(x – a)³ / 3! + … + f⁽ⁿ⁾(a)(x – a)ⁿ / n! + …

Here f⁽ⁿ⁾(a) denotes the n-th derivative of f evaluated at a. The Maclaurin series is obtained by setting a = 0, giving:

这里 f⁽ⁿ⁾(a) 表示 f 在 a 处的 n 阶导数。令 a = 0 即得麦克劳林级数：

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

Notice the alternating factorial denominators; n! grows very fast, which helps the series converge for many common functions.

请注意交替出现的阶乘分母；n! 增长极快，这有助于许多常见函数的级数收敛。

4. Taylor Polynomials | 泰勒多项式

In practice we rarely use an infinite series. Instead, we truncate after a few terms to obtain a Taylor polynomial. The polynomial of degree n, written Pₙ(x), approximates f(x) near x = a. As n increases, the approximation becomes more accurate. For instance, P₁(x) is the linear tangent approximation, while P₂(x) adds a quadratic curvature correction.

实际中我们很少使用无穷级数，而是截取前几项得到一个泰勒多项式。n 次多项式 Pₙ(x) 在 x = a 附近逼近 f(x)。随着 n 增加，逼近精度提高。例如，P₁(x) 是线性切线近似，而 P₂(x) 增加了一个二次曲率修正。

When you see questions asking for ‘the first three non-zero terms of the Maclaurin series’, you are being asked to write P₂(x) or P₄(x) depending on which terms vanish.

当你看到题目要求“写出麦克劳林级数的前三个非零项”时，实际上就是让你写出 P₂(x) 或 P₄(x)（取决于哪些项为零）。

5. Series for eˣ | eˣ 的级数展开

The exponential function eˣ has the simplest Maclaurin series because all its derivatives are eˣ and e⁰ = 1. This yields:

指数函数 eˣ 的麦克劳林级数最简单，因为它所有的导数都是 eˣ，且 e⁰ = 1。由此得到：

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

This expansion is valid for all real x, meaning the radius of convergence is infinite. You can use the first few terms to estimate e⁰.⁵ rapidly: 1 + 0.5 + 0.5²/2 + 0.5³/6 = 1 + 0.5 + 0.125 + 0.020833… = 1.64583, which is already close to the true value 1.64872.

该展开式对所有实数 x 都成立，即收敛半径为无穷大。你可以用前几项快速估算 e⁰·⁵：1 + 0.5 + 0.5²/2 + 0.5³/6 = 1 + 0.5 + 0.125 + 0.020833… = 1.64583，这与真实值 1.64872 已非常接近。

6. Series for sin x and cos x | 正弦与余弦函数的展开

The Maclaurin series for sin x and cos x alternate between zero and non-zero terms because derivatives cycle every four steps. All even derivatives of sin x vanish at 0; all odd derivatives of cos x vanish at 0. The expansions are:

sin x 和 cos x 的麦克劳林级数在零项和非零项之间交替，因为每四阶导数循环一次。sin x 的所有偶数阶导数在 0 处为零；cos x 的所有奇数阶导数在 0 处为零。展开式如下：

sin x = x – x³ / 3! + x⁵ / 5! – x⁷ / 7! + …

cos x = 1 – x² / 2! + x⁴ / 4! – x⁶ / 6! + …

These series also converge for all real x. The alternating signs appear because the derivatives at 0 cycle through 1, 0, -1, 0. A typical exam question might ask you to derive the first three non-zero terms of the Maclaurin expansion for sin(2x) by substituting 2x into the standard series.

这两个级数也对所有实数 x 收敛。符号交替出现是因为在 0 处的导数值循环为 1、0、-1、0。典型的考题可能会让你通过将 2x 代入标准级数来推导 sin(2x) 的麦克劳林展开式的前三个非零项。

7. Series for ln(1+x) | 自然对数 ln(1+x) 的展开

The function ln(1+x) does not have a Maclaurin expansion at x=0? It does, because ln(1+0)=0. Its derivatives yield a pattern: f'(x)=1/(1+x), f”(x)=-1/(1+x)², etc. Evaluating at 0 gives:

函数 ln(1+x) 在 x=0 处可以展开，因为 ln(1+0)=0。其导数呈现规律：f'(x)=1/(1+x), f”(x)=-1/(1+x)² 等。在 0 处估值得到：

ln(1+x) = x – x² / 2 + x³ / 3 – x⁴ / 4 + … + (-1)ⁿ⁺¹ xⁿ / n + …

This series converges only for -1 < x ≤ 1. The radius of convergence is 1. For x=1 you get the alternating harmonic series ln 2 = 1 - 1/2 + 1/3 - 1/4 + ….

该级数仅在 -1 < x ≤ 1 时收敛，收敛半径为 1。当 x=1 时，得到交错调和级数 ln 2 = 1 - 1/2 + 1/3 - 1/4 + …。

8. Binomial Expansion Connection | 与二项展开式的联系

The binomial expansion for (1+x)ᵏ with rational or negative k is essentially a Maclaurin series. The general formula is:

当 k 为有理数或负数时，(1+x)ᵏ 的二项式展开本质上就是麦克劳林级数。一般公式为：

(1+x)ᵏ = 1 + kx + k(k-1)x² / 2! + k(k-1)(k-2)x³ / 3! + …

This series is valid for |x| < 1. For positive integer k it truncates to the familiar finite binomial expansion. IGCSE students often use this as a shortcut to find series without differentiating repeatedly.

该级数在 |x| < 1 时成立。当 k 为正整数时，它截断为我们熟悉的有限二项展开式。IGCSE 学生常常将此作为捷径，无需反复求导即可得到级数。

9. Approximation and Error Estimation | 近似与误差估计

When we use a truncated Taylor polynomial, the error can be estimated using the Lagrange remainder term. For an expansion around a, the error after n terms is:

当我们使用截断的泰勒多项式时，可以用拉格朗日余项估计误差。对于围绕 a 点的展开，n 项之后的误差为：

Rₙ(x) = f⁽ⁿ⁺¹⁾(c) (x – a)ⁿ⁺¹ / (n+1)!

where c lies between a and x. In exam-style questions, you may be told that |f⁽ⁿ⁺¹⁾(c)| ≤ M and then asked to find the maximum error. Understanding this concept helps you decide how many terms are needed for a desired accuracy.

其中 c 介于 a 和 x 之间。在考试风格的题目中，你可能会被告知 |f⁽ⁿ⁺¹⁾(c)| ≤ M，然后要求找到最大误差。理解这个概念有助于你判断需要取多少项才能达到所需的精度。

10. Radius of Convergence | 收敛半径

Not all Taylor series converge for all x. The interval of convergence can be found using the ratio test. For instance, the Maclaurin series for 1/(1-x) is the geometric series 1 + x + x² + x³ + … which converges only for |x| < 1. The radius of convergence R is 1. Knowing R helps you avoid using a series outside its valid domain.

并非所有泰勒级数都对所有 x 收敛。收敛区间可用比值判别法求得。例如，1/(1-x) 的麦克劳林级数为几何级数 1 + x + x² + x³ + …，它仅在 |x| < 1 时收敛，收敛半径 R = 1。了解 R 有助于避免在无效域中使用级数。

11. Common Applications | 常见应用

Taylor series are not just abstract; they are used to compute numerical values of functions without a calculator, solve differential equations approximately, and evaluate complicated integrals by expanding the integrand. In IGCSE style, you might be asked to use the first three terms of eˣ to find an approximate solution to an equation like eˣ = 2x + 1 near x = 0.

泰勒级数并非抽象概念；它们被用于在无计算器时计算函数的数值、近似求解微分方程，以及通过展开被积函数来计算复杂积分。在 IGCSE 风格的题目中，你可能会被要求使用 eˣ 的前三项来近似求解方程，如 eˣ = 2x + 1 在 x = 0 附近的解。

Another classic application is finding limits: replacing sin x by x – x³/6 allows you to evaluate lim(x→0) (sin x – x)/x³ easily.

另一个经典应用是求极限：用 x – x³/6 替代 sin x 可以轻松计算 lim(x→0) (sin x – x)/x³。

12. Exam Tips | 考试技巧

When tackling Taylor series questions, always start by computing the first few derivatives at the given point. Look for patterns to avoid unnecessary work. If a question asks for the series up to the x³ term, you usually need derivatives up to the third order. Remember to include factorial denominators – a common mistake is to write x² instead of x²/2!.

在解答泰勒级数题目时，务必先计算给定点处的前几阶导数。寻找规律以避免不必要的工作。如果要求写出到 x³ 项的级数，通常需要算到三阶导数。一定要记得包含阶乘分母——一个常见错误是写成 x² 而不是 x²/2!。

Also, check whether the expansion point is a=0 (Maclaurin) or another value. Substituting back into the formula correctly is key. Finally, verify your series by testing a simple value, like x=0.1, to see if your approximation agrees with a calculator.

此外，检查展开点是 a=0（麦克劳林）还是其他值。正确代入公式至关重要。最后，通过测试一个简单值（如 x=0.1）来验证你的级数，看看近似值是否与计算器结果一致。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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