The Binomial Expansion | 二项式展开

📚 The Binomial Expansion | 二项式展开

The binomial expansion is one of the most important results in A-Level mathematics. It allows us to expand expressions of the form (a+b)ⁿ without laborious multiplication. More remarkably, with a generalised version, we can even expand (1+x)ⁿ for any rational index n, turning algebraic expressions into infinite series that approximate functions.

二项式展开是 A-Level 数学中最重要的结果之一。它使我们无需冗长的乘法即可展开形如 (a+b)ⁿ 的表达式。更奇妙的是，通过推广，我们还可以将任何有理数指数 n 的 (1+x)ⁿ 展开为无穷级数，从而逼近函数值。

1. The Binomial Theorem for Positive Integer n | 正整数 n 的二项式定理

For a positive integer n, the binomial theorem states that:

对正整数 n，二项式定理表述如下：

(a+b)ⁿ = Σₖ₌₀ⁿ ⁿCₖ aⁿ⁻ᵏ bᵏ

where ⁿCₖ = n! / (k!(n−k)!) is called the binomial coefficient. This gives a finite expansion with exactly n+1 terms.

其中 ⁿCₖ = n! / (k!(n−k)!) 称为二项式系数。该展开是有限的，共有 n+1 项。

2. Factorial Notation | 阶乘记号

Factorials are defined as n! = n × (n−1) × … × 2 × 1, with 0! = 1 by convention. They simplify many algebraic expressions and are essential for computing binomial coefficients.

阶乘定义为 n! = n × (n−1) × … × 2 × 1，并规定 0! = 1。阶乘简化了许多代数表达式，也是计算二项式系数的基础。

ⁿCₖ = ⁿ! / (k!(n−k)!), ⁴C₂ = 4! / (2! 2!) = 6

Notice that ⁿCₖ = ⁿCₙ₋ₖ, a symmetry that often saves time.

注意 ⁿCₖ = ⁿCₙ₋ₖ，这个对称性常可节省计算时间。

3. Pascal’s Triangle | 杨辉三角

Pascal’s triangle provides a quick way to generate binomial coefficients for small n. Each row begins and ends with 1, and each interior number is the sum of the two numbers above it.

杨辉三角为较小的 n 提供了一种快速生成二项式系数的方法。每一行以 1 开始和结束，而每个内部数字等于其上方两个数字之和。

1
1	1
1	2	1
1	3	3	1
1	4	6	4	1

Using Pascal’s triangle, we can immediately write (1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴.

利用杨辉三角，我们可以直接写出 (1+x)⁴ = 1 + 4x + 6x² + 4x³ + x⁴。

4. The General Term | 通项

In the expansion of (a+b)ⁿ, the term involving bᵏ is given by Tₖ₊₁ = ⁿCₖ aⁿ⁻ᵏ bᵏ. This formula is useful when we need a particular term without writing out the whole expansion.

在 (a+b)ⁿ 的展开中，含 bᵏ 的项为 Tₖ₊₁ = ⁿCₖ aⁿ⁻ᵏ bᵏ。当我们需要特定某项而无需写出整个展开时，这个公式很有用。

Tₖ₊₁ = ⁿCₖ aⁿ⁻ᵏ bᵏ

For example, to find the x³ term in (2+3x)⁵, set k=3: T₄ = ⁵C₃ (2)² (3x)³ = 10 × 4 × 27x³ = 1080x³.

例如，要求 (2+3x)⁵ 中 x³ 项的系数，令 k=3：T₄ = ⁵C₃ (2)² (3x)³ = 10 × 4 × 27x³ = 1080x³。

5. Binomial Coefficients and Their Properties | 二项式系数及其性质

The coefficients ⁿCₖ are integers with several important properties:

系数 ⁿCₖ 是整数，具有几个重要性质：

Symmetry: ⁿCₖ = ⁿCₙ₋ₖ
Addition rule: ⁿCₖ + ⁿCₖ₊₁ = ⁿ⁺¹Cₖ₊₁
Sum of coefficients: Σₖ₌₀ⁿ ⁿCₖ = 2ⁿ

These properties are used in proofs, probability, and algebra.

这些性质在证明、概率和代数中都有应用。

6. The Binomial Expansion for Rational n | 有理数指数 n 的展开

The true power of the binomial expansion emerges when we replace the positive integer n with any rational number n. For |x|<1 we can write:

当我们将正整数 n 替换为任意有理数 n 时，二项式展开的真正威力才显现出来。对于 |x|<1，我们可以写成：

(1+x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …

This is now an infinite series unless n is a non-negative integer. It converges only when |x|<1 (or in some special cases).

除非 n 是非负整数，否则这现在是一个无穷级数。仅当 |x|<1（或某些特殊情况）时它才收敛。

7. The Condition for Convergence |x|<1 | 收敛条件 |x|<1

The series for (1+x)ⁿ with non-integer n converges absolutely if |x|<1. The proof involves ratio tests from pure mathematics. In practice, AQA exam questions always specify the range of validity.

对于非整数 n 的 (1+x)ⁿ 级数，当 |x|<1 时绝对收敛。证明要用到纯数学中的比值审敛法。在实际考试中，AQA 试题总会明确给出的有效范围。

|x| < 1

For example, √(1+x) = (1+x)^½ = 1 + ½x − ⅛x² + … is valid for −1 < x < 1.

例如，√(1+x) = (1+x)^½ = 1 + ½x − ⅛x² + … 在 −1 < x < 1 时有效。

8. Using Partial Fractions with Binomial Expansion | 结合部分分式

When expanding a rational expression, we first decompose it into partial fractions, then apply the binomial expansion to each part individually.

当展开有理式时，我们先将其分解为部分分式，然后分别对每一部分应用二项式展开。

Suppose f(x) = 1/((1−2x)(1+x)). Write f(x) = A/(1−2x) + B/(1+x). Solving gives A = 2/3, B = 1/3. Then:

设 f(x) = 1/((1−2x)(1+x))。写出 f(x) = A/(1−2x) + B/(1+x)。解得 A = 2/3, B = 1/3。于是：

(2/3)(1−2x)⁻¹ + (1/3)(1+x)⁻¹

Each term can now be expanded using (1+u)⁻¹ = 1 − u + u² − u³ + … with an appropriate u.

现在可以分别利用 (1+u)⁻¹ = 1 − u + u² − u³ + …（取合适的 u）展开每一项。

9. Applications in Approximations | 在近似计算中的应用

The binomial expansion allows us to approximate powers and roots with great efficiency. For instance, to approximate √1.05, set x=0.05 in (1+x)^½:

二项式展开使我们能够高效地近似计算乘方与根式。例如，要近似 √1.05，在 (1+x)^½ 中令 x=0.05：

1 + ½(0.05) − ⅛(0.05)² ≈ 1.0246875

Compare with the calculator value 1.024695 – an error of less than 10⁻⁵. This method underpins many numerical algorithms.

与计算器所得 1.024695 相比，误差小于 10⁻⁵。这一方法构成了许多数值算法的基础。

10. Common Mistakes and Exam Tips | 常见错误与应试技巧

Students often forget the validity condition or misapply the general formula when n is negative or fractional. Here are crucial tips:

学生经常忘记有效性条件，或在 n 为负数或分数时错用通项公式。以下是一些关键提示：

Always check that |x|<1 before using the infinite binomial expansion.
Write (a+bx)ⁿ as aⁿ(1 + (b/a)x)ⁿ to use the standard form.
For partial fractions, ensure the numerator degree is lower than the denominator.
When finding a particular term, use the general term formula carefully with k starting at 0.
State the range of validity explicitly in your answer.

中文翻译：始终检查是否 |x|<1 再使用无穷二项展开；将 (a+bx)ⁿ 写成 aⁿ(1 + (b/a)x)ⁿ 以利用标准形式；使用部分分式时要确保分子次数低于分母；求特定项时注意从 k=0 开始正确使用通项公式；在答案中明确写出有效范围。

In summary, the binomial expansion is a cornerstone of A-Level mathematics. Mastering both the finite and infinite versions gives you a powerful toolkit for algebra, series, and approximation. Practice the convergence conditions and the interplay with partial fractions to secure full marks in exam questions.

总之，二项式展开是 A-Level 数学的基石。掌握有限与无限两种形式，就拥有了解代数、级数和近似计算的强大工具。务必练习收敛条件以及与部分分式的结合使用，从而在考试中稳定获得满分。

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The Binomial Expansion | 二项式展开