Maclaurin Series Expansion | 麦克劳林级数展开 考点精讲

📚 Maclaurin Series Expansion | 麦克劳林级数展开 考点精讲

The Maclaurin series is a fundamental tool in IGCSE Additional Mathematics that expresses functions as infinite polynomials. It is the Taylor series centred at x = 0, and mastering it allows you to approximate functions, evaluate limits, and solve exam problems efficiently. This article covers the essential formulas, derivations, standard series, common pitfalls, and exam strategies to help you excel.

麦克劳林级数是 IGCSE 附加数学中一个核心工具，它将函数表示为无穷多项式。它是中心在 x = 0 的泰勒级数，掌握它有助于你近似函数、计算极限、高效解决考试问题。本文涵盖基本公式、推导过程、标准级数、常见错误和应考策略，助你斩获高分。

1. What is Maclaurin Expansion? | 什么是麦克劳林展开？

A Maclaurin expansion (or Maclaurin series) is a power series representation of a function f(x) about x = 0. If f(x) is infinitely differentiable at 0, its Maclaurin series is the sum of terms f⁽ⁿ⁾(0)xⁿ/n! from n = 0 to infinity. It is an invaluable approximation tool because truncating the series yields a polynomial that approximates the function near x = 0.

麦克劳林展开（又称麦克劳林级数）是函数 f(x) 在 x = 0 附近的幂级数表示。若 f(x) 在 0 处无限次可导，其麦克劳林级数即为 n=0 到无穷的 f⁽ⁿ⁾(0)xⁿ/n! 之和。由于截断级数可得到在 x = 0 附近近似函数的若干项多项式，因此它是一种非常实用的近似工具。

2. The Maclaurin Series Formula and Standard Expansions | 麦克劳林级数公式与标准展开式

The general Maclaurin series for a function f(x) is:

函数 f(x) 的通用麦克劳林级数为：

f(x) = f(0) + f′(0)x + f″(0)/2! x² + f‴(0)/3! x³ + … + f⁽ⁿ⁾(0)/n! xⁿ + …

Here, f′(0), f″(0), f‴(0), … denote the first, second, third, … derivatives evaluated at x = 0. The expression n! (n factorial) appears in the denominator of each term. The series may converge for all real x or only within a radius of convergence.

其中 f′(0)、f″(0)、f‴(0) … 表示在 x = 0 处的一阶、二阶、三阶 … 导数值。每一项的分母均含有 n!（n 的阶乘）。该级数可能对所有实数 x 收敛，也可能仅在一个收敛半径内成立。

The following standard Maclaurin series are essential for IGCSE exams. Memorise them – they will save you valuable time in deriving expansions and solving approximation problems.

以下标准麦克劳林级数是 IGCSE 考试的必背内容。熟记它们，可为你推导展开式和求解近似问题节省宝贵时间。

• eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + … (valid for all real x) | eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + … （对所有实数成立）
• sin x = x − x³/3! + x⁵/5! − x⁷/7! + … (valid for all x) | sin x = x − x³/3! + x⁵/5! − x⁷/7! + … （对所有实数成立）
• cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … (valid for all x) | cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + … （对所有实数成立）
• ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + x⁵/5 − … (valid for −1 < x ≤ 1) | ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + x⁵/5 − … （成立范围 −1 < x ≤ 1）
• (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … (valid for |x| < 1, any real n) | (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … （成立范围 |x| < 1，n 为任意实数）

3. Deriving the Expansion of eˣ | 推导 eˣ 的展开

To derive the Maclaurin series for eˣ, let f(x) = eˣ. The beauty of the exponential function is that all its derivatives are eˣ. Thus f(0) = 1, f′(0) = 1, f″(0) = 1, and in general f⁽ⁿ⁾(0) = 1 for every n. Substituting into the general formula yields:

为推导 eˣ 的麦克劳林级数，令 f(x) = eˣ。指数函数的妙处在于其各阶导数仍为 eˣ。因此 f(0) = 1，f′(0) = 1，f″(0) = 1，且一般地 f⁽ⁿ⁾(0) = 1 对所有 n 成立。代入通用公式即得：

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

This series converges for every real number x, which is why it is so useful. You can verify that differentiating the right-hand side reproduces the series itself, reflecting the property (eˣ)′ = eˣ.

该级数对所有实数 x 都收敛，这也是它用途广泛的原因。你可以验证，对右端逐项求导会重现此级数，这正反映了 (eˣ)′ = eˣ 的性质。

4. Deriving the Expansions of sin x and cos x | 推导 sin x 与 cos x 的展开

For f(x) = sin x, compute successive derivatives: f′(x) = cos x, f″(x) = −sin x, f‴(x) = −cos x, f⁗(x) = sin x, and then the pattern repeats every four derivatives. Evaluating at x = 0 gives: f(0) = 0, f′(0) = 1, f″(0) = 0, f‴(0) = −1, f⁗(0) = 0, etc. Only odd-order derivatives produce non‑zero values at 0. Plugging these into the formula results in:

对于 f(x) = sin x，逐次求导得：f′(x) = cos x，f″(x) = −sin x，f‴(x) = −cos x，f⁗(x) = sin x，此后每四阶循环一次。代入 x = 0 得：f(0) = 0，f′(0) = 1，f″(0) = 0，f‴(0) = −1，f⁗(0) = 0 等。只有奇数阶导数在 0 处非零。代入公式可得：

sin x = x − x³/3! + x⁵/5! − x⁷/7! + …

For cos x, a similar analysis shows: f(0) = 1, f′(0) = 0, f″(0) = −1, f‴(0) = 0, f⁗(0) = 1, … producing only even powers:

对于 cos x，类似分析表明：f(0) = 1，f′(0) = 0，f″(0) = −1，f‴(0) = 0，f⁗(0) = 1，… 仅出现偶次幂：

cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …

Notice how the sine series contains only odd powers with alternating signs, while the cosine series contains only even powers with alternating signs. These patterns are

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