Maclaurin Series: Expansion and Applications | 麦克劳林级数的展开与应用

📚 Maclaurin Series: Expansion and Applications | 麦克劳林级数的展开与应用

The Maclaurin series is a powerful tool in IB Mathematics that allows us to represent a wide variety of functions as infinite polynomials. By expanding a function around x = 0, we can approximate values, evaluate limits, solve differential equations, and even derive elegant identities such as Euler’s formula. This article provides a structured, exam-focused guide to the expansion and application of Maclaurin series for IB students.

麦克劳林级数是IB数学中一个强大的工具，它使我们能够将多种函数表示为无穷多项式。通过在 x = 0 附近展开函数，我们可以近似计算函数值、求解极限、解微分方程，甚至推导出如欧拉公式这样优美的恒等式。本文为IB学生提供一份结构化、紧扣考点的麦克劳林级数展开与应用指南。

1. Definition of the Maclaurin Series | 麦克劳林级数的定义

For a function f(x) that is infinitely differentiable at x = 0, the Maclaurin series is given by:

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

More compactly, using summation notation:

f(x) = Σₙ₌₀^∞ f⁽ⁿ⁾(0) xⁿ / n!

Here f⁽ⁿ⁾(0) denotes the nth derivative of f evaluated at x = 0, and n! is the factorial of n. The series is named after Colin Maclaurin, though it is a special case of the Taylor series centred at a = 0.

对于一个在 x = 0 处无穷可微的函数 f(x)，其麦克劳林级数定义为：

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

更紧凑地，使用求和记号可写为：

f(x) = Σₙ₌₀^∞ f⁽ⁿ⁾(0) xⁿ / n!

其中 f⁽ⁿ⁾(0) 表示 f 的 n 阶导数在 x = 0 处的值，n! 是 n 的阶乘。该级数以科林·麦克劳林命名，实际上它是泰勒级数在 a = 0 时的特例。

2. Maclaurin Series for Common Functions | 常见函数的麦克劳林级数

The following series are essential for IB examinations. Memorising them, along with their intervals of convergence, saves time and reduces errors.

以下级数在IB考试中至关重要。牢记这些级数及其收敛区间，可以节省时间并减少错误。

eˣ = 1 + x + x²/2! + x³/3! + … for all real x
sin x = x − x³/3! + x⁵/5! − x⁷/7! + … for all real x
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … for all real x
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … for −1 < x ≤ 1
(1+x)ᵖ = 1 + px + p(p−1)x²/2! + p(p−1)(p−2)x³/3! + … for |x| < 1 (binomial series)

对于任意实数 x：eˣ = 1 + x + x²/2! + x³/3! + …；sin x = x − x³/3! + x⁵/5! − x⁷/7! + …；cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …。对于 −1 < x ≤ 1：ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + …。对于 |x| < 1（二项式级数）：(1+x)ᵖ = 1 + px + p(p−1)x²/2! + p(p−1)(p−2)x³/3! + …。

tan⁻¹x = x − x³/3 + x⁵/5 − x⁷/7 + … for |x| ≤ 1

The arctangent series is particularly useful when deriving π approximations such as the Gregory–Leibniz series.

反正切级数在推导π的近似值（如格雷戈里–莱布尼茨级数）时特别有用。

3. How to Derive a Maclaurin Expansion | 如何推导麦克劳林展开式

To expand a function from first principles, follow these steps:

要从基本原理展开一个函数，请遵循以下步骤：

Differentiate the function repeatedly and evaluate each derivative at x = 0.
Substitute f(0), f'(0), f”(0), … into the Maclaurin formula.
Simplify and look for a pattern, writing the series in sigma notation if possible.

反复求导，并计算各阶导数在 x = 0 处的值。
将 f(0)、f'(0)、f”(0)… 代入麦克劳林公式。
化简并寻找规律，尽可能用求和记号写出级数。

For example, let f(x) = sin x. Then f(0) = 0, f'(0) = 1, f”(0) = 0, f”'(0) = −1, and the pattern repeats every four derivatives. Substituting into the formula gives the familiar alternating series.

例如，设 f(x) = sin x。则 f(0) = 0，f'(0) = 1，f”(0) = 0，f”'(0) = −1，且每四阶导数循环一次。代入公式即可得到熟悉的交错级数。

4. Convergence and Remainder Term | 收敛性与余项

A Maclaurin series may converge only for certain values of x. The interval of convergence is often found using the ratio test. For example, the series for ln(1+x) converges for −1 < x ≤ 1, while the series for (1+x)ᵖ converges for |x| < 1 (with endpoint behaviour depending on p).

麦克劳林级数可能只在某些 x 值下收敛。通常使用比值检验来求收敛区间。例如，ln(1+x) 的级数在 −1 < x ≤ 1 上收敛，而 (1+x)ᵖ 的级数在 |x| < 1 上收敛（端点情况取决于 p）。

The remainder term Rₙ(x) measures the error when truncating the series after the nth-degree term. One form is the Lagrange remainder:

余项 Rₙ(x) 衡量截断到 n 次项时的误差。拉格朗日余项的一种形式为：

Rₙ(x) = f⁽ⁿ⁺¹⁾(c) xⁿ⁺¹ / (n+1)! for some c between 0 and x.

For IB, you are usually asked to show that the remainder tends to 0 as n → ∞ for a given interval, or to bound the error for a specific approximation.

在IB中，通常要求证明在给定区间上余项随 n → ∞ 趋于0，或者估计某个近似值的误差范围。

5. Approximation of Function Values | 函数值的近似计算

One of the most direct applications is approximating values such as sin 0.1, e⁻², or ln 1.2 using the first few terms. The more terms you include, the more accurate the approximation becomes.

最直接的应用之一是用前几项近似计算如 sin 0.1、e⁻² 或 ln 1.2 等值。项数越多，近似越精确。

For example, take eˣ with x = 0.5. Using the first four terms:

例如，取 eˣ 在 x = 0.5 处。使用前四项：

e⁰·⁵ ≈ 1 + 0.5 + 0.5²/2! + 0.5³/3! = 1 + 0.5 + 0.125 + 0.020833 = 1.645833

The actual value is 1.64872, so the error is about 0.00289. Including the fourth-order term reduces the error further.

实际值为 1.64872，因此误差约为 0.00289。加上四阶项后误差会更小。

When using a series for approximation, always check that the value of x lies within the interval of convergence. If x is too large, the truncated series may give a poor approximation.

使用级数做近似时，务必检查 x 的值是否落在收敛区间内。如果 x 太大，截断级数可能给出较差的近似。

6. Evaluating Limits Using Maclaurin Series | 用麦克劳林级数求极限

Maclaurin series provide an elegant alternative to L’Hôpital’s rule for limits involving indeterminate forms. By replacing functions with their polynomial expansions, many limits simplify to direct substitution.

对于涉及不定式的极限，麦克劳林级数提供了洛必达法则之外的一种优雅方法。将函数替换为多项式展开后，许多极限可以直接代入求得。

Example: Evaluate limₓ→₀ (sin x − x)/x³.

例：求 limₓ→₀ (sin x − x)/x³。

Using sin x = x − x³/3! + x⁵/5! − …, we have:

利用 sin x = x − x³/3! + x⁵/5! − …，可得：

(sin x − x)/x³ = ( − x³/3! + x⁵/5! − … ) / x³ = −1/6 + x²/5! − …

As x → 0, all higher-order terms vanish, so the limit is −1/6. This method is often quicker than repeated differentiation.

当 x → 0 时，所有高阶项消失，因此极限为 −1/6。这种方法往往比反复求导更快。

7. Integration and Differential Equations | 积分与微分方程

Some functions, such as e^(−x²) or sin(x²), do not have elementary antiderivatives. Maclaurin series allow us to integrate such functions term by term, giving a polynomial approximation of the definite integral.

某些函数如 e^(−x²) 或 sin(x²) 没有初等原函数。麦克劳林级数允许我们逐项积分，从而得到定积分的多项式近似。

Example: Approximate ∫₀¹ e^(−x²) dx using the first three terms.

例：用前三项近似 ∫₀¹ e^(−x²) dx。

Since e^u = 1 + u + u²/2! + …, substitute u = −x²:

因为 e^u = 1 + u + u²/2! + …，令 u = −x²：

e^(−x²) ≈ 1 − x² + x⁴/2

Integrating from 0 to 1 gives 1 − 1/3 + 1/10 = 0.7667 (compared with the actual 0.7468). More terms improve accuracy.

从0到1积分得到 1 − 1/3 + 1/10 = 0.7667（实际值为 0.7468）。项数越多精度越高。

Maclaurin series can also solve simple differential equations near x = 0. Assume y = Σ aₙxⁿ, substitute into the equation, and equate coefficients to find the recurrence relation for aₙ.

麦克劳林级数还可以求解 x = 0 附近的简单微分方程。设 y = Σ aₙxⁿ，代入方程并比较系数，即可得到 aₙ 的递推关系。

8. Euler’s Formula and Complex Numbers | 欧拉公式与复数

A famous application of Maclaurin series is deriving Euler’s formula. Expanding eⁱˣ, where i = √(−1), and separating real and imaginary parts gives:

麦克劳林级数的一个著名应用是推导欧拉公式。展开 eⁱˣ，其中 i = √(−1)，并将实部与虚部分开：

eⁱˣ = cos x + i sin x

When x = π, this yields eⁱᵖⁱ + 1 = 0, the identity that links five fundamental constants.

当 x = π 时，得到 eⁱᵖⁱ + 1 = 0，这个恒等式将五个基本常数联系在一起。

The derivation uses the alternating patterns in the series for sin x and cos x. This connection is frequently tested in IB Paper 3 or in questions on complex numbers and series.

该推导利用了 sin x 和 cos x 级数中的交错模式。这一联系在IB Paper 3或涉及复数与级数的题目中经常出现。

9. Products and Compositions of Functions | 函数乘积与复合的展开

Sometimes the required series is not a standard form but can be obtained by multiplying or composing known series. For example, to expand eˣ sin x, multiply the first few terms of each series and collect like powers of x.

有时所需的级数并不是标准形式，但可以通过已知级数的乘法或复合得到。例如，展开 eˣ sin x 时，将两个级数的前几项相乘并合并 x 的同幂次项即可。

Example: Find the Maclaurin series up to x³ for eˣ sin x.

例：求 eˣ sin x 到 x³ 项的麦克劳林级数。

eˣ = 1 + x + x²/2 + x³/6 + … and sin x = x − x³/6 + … Multiplying:

eˣ = 1 + x + x²/2 + x³/6 + …，sin x = x − x³/6 + …。相乘：

eˣ sin x = x + x² + (1/2 − 1/6)x³ + … = x + x² + x³/3 + …

This technique is useful when the function is a product of two or more standard functions.

当函数是两个或多个标准函数的乘积时，这一技巧非常有用。

10. The Binomial Series and Its Uses | 二项式级数及其应用

The general binomial expansion (1+x)ᵖ = Σₙ₌₀^∞ C(p, n) xⁿ, where C(p, n) = p(p−1)…(p−n+1)/n!, extends the binomial theorem to any real exponent p. It is a special Maclaurin series and is essential for expanding expressions such as √(1+x) or 1/(1+x).

一般二项式展开 (1+x)ᵖ = Σₙ₌₀^∞ C(p, n) xⁿ，其中 C(p, n) = p(p−1)…(p−n+1)/n!，将二项式定理推广到任意实数指数 p。它是麦克劳林级数的一个特例，对展开如 √(1+x) 或 1/(1+x) 的表达式至关重要。

For example, (1+x)⁻¹ = 1 − x + x² − x³ + … for |x| < 1, which is the geometric series. One common IB question is to expand √(4+x) by writing it as 2(1 + x/4)¹ᐟ² and then using the binomial series with p = 1/2.

例如，(1+x)⁻¹ = 1 − x + x² − x³ + …，|x| < 1，即几何级数。一个常见的IB题是将 √(4+x) 写成 2(1 + x/4)¹ᐟ²，然后使用 p = 1/2 的二项式级数展开。

11. Exam Tips and Common Pitfalls | 考试技巧与常见易错点

IB students frequently lose marks on Maclaurin series questions due to a few recurring errors. Being aware of these will help you avoid them.

IB学生在麦克劳林级数题目中常因一些反复出现的错误而失分。了解这些错误有助于避免。

Pitfall \| 易错点	Advice \| 建议
Forgetting factorials in the denominators of terms beyond x².	Always write out the first few derivatives and check the formula f⁽ⁿ⁾(0) xⁿ/n!.
Using a series outside its interval of convergence.	State the interval of convergence when asked, and check that the x value lies inside it before approximating.
Confusing Maclaurin series with Taylor series centred at other points.	Remember: Maclaurin series is a Taylor series at x = 0.
Incorrect signs in alternating series.	Check the sign of each derivative at 0; for sin x, signs alternate: +, −, +, −…
Dropping higher-order terms too early in limit evaluations.	Keep enough terms to cancel the lowest-order denominator correctly.

In exam questions, read carefully whether the question asks for the first three non-zero terms, the general term, or a coefficient of a specific power of x. Pay attention to the command term: ‘find’, ‘show’, ‘hence’ or ‘deduce’.

考试时请仔细阅读题目，是要求写出前三个非零项、通项，还是 x 的某次幂的系数。注意指令词：’find’（求）、’show’（证明）、’hence’（由此）或’deduce’（推断）。

12. Summary | 总结

The Maclaurin series is a cornerstone of mathematical analysis and a recurring topic in IB examinations. Mastery of standard expansions, convergence conditions, and application techniques—approximation, limits, integration, and differential equations—will give you a significant advantage in both Paper 2 and Paper 3.

麦克劳林级数是数学分析的基石，也是IB考试中的常考主题。熟练掌握标准展开式、收敛条件以及近似、极限、积分和微分方程等应用技巧，将在Paper 2和Paper 3中为你带来显著优势。

Practice by deriving the common series from scratch, solving past-paper questions, and always verifying that your results are reasonable. With consistent practice, Maclaurin series will become one of the most rewarding topics in your IB mathematics toolkit.

请通过从零推导常见级数、练习历年真题并始终检查结果是否合理来进行学习。在持续练习下，麦克劳林级数将成为你IB数学工具箱中最有价值的内容之一。

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Maclaurin Series: Expansion and Applications | 麦克劳林级数的展开与应用