📚 Example 4.2.3: Binomial Expansion of (1+2x)1/2 | 例4.2.3：(1+2x)1/2的二项展开式
In this AQA A-Level Pure Mathematics worked example, we use the general binomial expansion to expand (1+2x)1/2 in ascending powers of x up to the term in x3, state the range of validity, and use the expansion to estimate the square root of 1.02 numerically.
在本 AQA A-Level 纯数学例题中，我们使用一般二项展开式将 (1+2x)1/2 按 x 的升幂展开至 x³ 项，说明展开式成立的范围，并利用该展开式对根号 1.02 进行数值估算。
1. The Worked Example Statement | 例题陈述
Expand (1+2x)1/2 in ascending powers of x up to and including the term in x3. State the values of x for which the expansion is valid. Hence estimate the value of √1.02 to five decimal places.
将 (1+2x)1/2 按 x 的升幂展开，直到并包括 x³ 项。说明展开式成立的 x 取值范围。由此估算 √1.02 的值，结果保留五位小数。
This example tests two AQA Assessment Objectives: accurate use of the rational-power binomial formula and correct substitution into a finite series for approximation.
本题考查 AQA 两个评价目标：准确使用有理数幂的二项展开公式，以及正确代入有限级数进行近似计算。
2. General Binomial Expansion for Rational Powers | 有理数幂的一般二项展开式
For any real power n and |u| < 1, the general binomial expansion is given by the following infinite series.
对于任意实数幂 n，且 |u| < 1，一般二项展开式由以下无穷级数给出。
(1 + u)ⁿ = 1 + n u + n(n − 1)/2! u² + n(n − 1)(n − 2)/3! u³ + … , |u| < 1
Unlike the ordinary binomial theorem for positive integer powers, this series does not terminate. AQA questions normally require only the first few terms, usually up to x², x³ or x⁴.
与正整数幂的一般二项式定理不同，这个级数不会终止。AQA 考题通常只要求前几项，一般到 x²、x³ 或 x⁴ 项。
3. Substituting n = 1/2 and u = 2x | 代入 n = 1/2 与 u = 2x
xpression is u = 2x. Writing the expansion with these values gives the starting structure.
在本例中，幂为 n = 1/2，内层表达式为 u = 2x。将这些值代入展开式，得到初始结构。
(1 + 2x)1/2 = 1 + (1/2)(2x) + (1/2)(1/2 − 1)/2! (2x)² + (1/2)(1/2 − 1)(1/2 − 2)/3! (2x)³ + …
ackets when raising to powers. Many candidates incorrectly square only x instead of squaring the whole term 2x.
将 u = 2x 放到括号内后再进行乘方非常重要。许多考生只对 x 平方，而错误地没有对整个 2x 项平方。
4. Calculating the Coefficients | 系数计算
The first four coefficients can be calculated step by step. The table below summarises the working for the terms up to u³.
前四项系数可以逐步计算。下表总结了直到 u³ 各项的计算过程。
| Power of u | Coefficient calculation | Simplified term |
|---|---|---|
| u¹ | (1/2)(2x) | +x |
| u² | (1/2)(−1/2)/2! × (2x)² | −1/2 x² |
| u³ | (1/2)(−1/2)(−3/2)/3! × (2x)³ | +1/2 x³ |
For the u² term, the product (1/2)(−1/2) gives −1/4, then division by 2! gives −1/8. Multiplying by (2x)² = 4x² produces −1/2 x².
对于 u² 项，(1/2)(−1/2) 得到 −1/4，再除以 2! 得到 −1/8。乘以 (2x)² = 4x² 后得到 −1/2 x²。
For the u³ term, the product (1/2)(−1/2)(−3/2) equals 3/8. Dividing by 3! = 6 gives 1/16, and multiplying by (2x)³ = 8x³ gives +1/2 x³.
对于 u³ 项，(1/2)(−1/2)(−3/2) 等于 3/8。除以 3! = 6 得到 1/16，再乘以 (2x)³ = 8x³ 得到 +1/2 x³。
5. Simplified Expansion up to x³ | 化简至 x³ 的展开式
Substituting the simplified coefficients back into the series gives the required expansion up to the term in x³.
将化简后的系数代回级数，得到直至 x³ 项的展开式。
(1 + 2x)1/2 = 1 + x − 1/2 x² + 1/2 x³ + … , for |2x| < 1
The next term is −5/8 x⁴, which is useful to check accuracy in approximation questions. AQA often expects candidates to recognise that omitted terms are small but not zero.
下一项为 −5/8 x⁴，在近似计算问题中可用于检查精度。AQA 通常希望考生认识到省略的项虽小，但不为零。
6. Range of Validity | 展开式的有效范围
The general binomial expansion is valid when |u| < 1. Since u = 2x, this gives the condition |2x| < 1.
一般二项展开式在 |u| < 1 时成立。由于 u = 2x，因此得到 |2x| < 1。
|2x| < 1 ⇒ |x| < 1/2
So the expansion is valid for −1/2 < x < 1/2. AQA mark schemes require this inequality to be stated explicitly, often as a range rather than just |x| < 1/2.
因此展开式
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