📚 Binomial Estimation | 二项式估计
Binomial estimation is a powerful technique in A-Level Mathematics that uses the binomial expansion to approximate numerical values. When the power is not a positive integer, the binomial series provides an infinite polynomial that can be truncated to give a very accurate approximation for suitable values of x.
二项式估算是 A-Level 数学中一项至关重要的技巧，它利用二项展开式来近似计算数值。当幂次不是正整数时，二项级数构成一个无穷多项式；在合适的 x 取值下，截取前几项即可得到非常精确的近似值。
1. The Binomial Theorem for Positive Integers | 正整数指数的二项式定理
For a positive integer n, the binomial expansion is exact and finite:
对于正整数 n，二项展开是精确且有限的：
(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹b + ⁿC₂ aⁿ⁻²b² + … + bⁿ
eries. For estimation, however, we usually need the case where a = 1, written as (1 + x)ⁿ.
这个常见形式是通用二项级数的基础。然而在估算中，我们通常需要 a = 1 的情形，即 (1 + x)ⁿ。
2. Extending to Rational and Negative Powers | 推广到有理数和负数指数
For any rational number n, provided |x| < 1, the expansion becomes an infinite series:
对于任意有理数 n，只要满足 |x| < 1，展开式就变成一个无穷级数：
(1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + …
This expansion is valid when n is negative or fractional, and it is the core of binomial estimation.
当 n 为负数或分数时，该展开式依然有效，这正是二项式估算的核心。
3. The Convergent Condition | 收敛条件
The condition |x| < 1 guarantees that the infinite series converges. For small values of x, the terms decrease rapidly, so keeping only the first few terms yields a good approximation.
条件 |x| < 1 保证了无穷级数收敛。当 x 很小时，各项递减很快，因此只保留前几项就能得到良好的近似。
- If |x| is close to 1, more terms are needed for the same accuracy.
- 如果 |x| 接近 1，要达到相同精度需要保留更多项。
- If |x| is very small (e.g., 0.01), two or three terms are often enough.
- 如果 |x| 很小（如 0.01），通常两到三项就足够了。
4. Rearranging Expressions | 表达式的变形
Many expressions are not given as (1 + x)ⁿ directly. To apply binomial estimation, rewrite them in the form aⁿ(1 + b/a)ⁿ, ensuring |b/a| < 1.
许多表达式并不是直接以 (1 + x)ⁿ 的形式给出。为了应用二项式估算，我们需要将其改写成 aⁿ(1 + b/a)ⁿ，并确保 |b/a| < 1。
For example, √17 = √16 + 1 = 4(1 + 1/16)1/2. Here b/a = 1/16, which is much less than 1.
例如，√17 = √16 + 1 = 4(1 + 1/16)1/2。这里 b/a = 1/16，远小于 1。
5. Estimation Steps | 估算步骤
Follow these systematic steps when performing binomial estimation:
进行二项式估算时，请遵循以下系统步骤：
- Identify n and the constant a so the expression can be written as aⁿ(1 + x)ⁿ.
- 确定 n 和常数 a，使表达式可以写成 aⁿ(1 + x)ⁿ。
- Ensure |x| < 1.
- 确保 |x| < 1。
- Expand the first few terms of (1 + x)ⁿ using the binomial series.
- 使用二项级数展开 (1 + x)ⁿ 的前几项。
- Substitute the chosen small value of x.
- 代入所选定的小 x 值。
- Calculate the result and, if required, state the degree of accuracy.
- 计算结果，若需要则说明精确程度。
6. Worked Example 1 | 例题一
Expand (1 + 2x)1/2 up to the term in x², and hence estimate √1.02.
将 (1 + 2x)1/2 展开到 x² 项，并据此估算 √1.02。
(1 + 2x)1/2 = 1 + (1/2)(2x) + (1/2)(−1/2)/2! (2x)² + …
Simplifying gives:
化简得到：
1 + x − ½x² + …
To estimate √1.02, write √1.02 = (1 + 0.02)1/2. Set 2x = 0.02, so x = 0.01. Then:
为了估算 √1.02，我们写 √1.02 = (1 + 0.02)1/2。令 2x = 0.02，即 x = 0.01，则：
√1.02 ≈ 1 + 0.01 − ½(0.01)² = 1.00995
This is accurate to at least five decimal places.
该结果至少精确到小数点后五位。
7. Worked Example 2 | 例题二
Use the binomial expansion to estimate √101 correct to four decimal places.
利用二项展开式估算 √101，精确到四位小数。
First, rewrite √101 as:
首先，将 √101 改写为：
√101 = 10√1 + 1/100 = 10(1 + 0.01)1/2
Using the expansion (1 + x)1/2 with x = 0.01:
使用 (1 + x)1/2 的展开式，令 x = 0.01：
(1 + 0.01)1/2 ≈ 1 + 0.01/2 − (0.01)²/8 + (0.01)³/16
≈ 1 + 0.005 − 0.0000125 + 0.0000000625 = 1.0049875625
Multiplying by 10 gives √101 ≈ 10.049875625, which rounds to 10.0499.
乘以 10 得到 √101 ≈ 10.049875625，四舍五入为 10.0499。
8. Accuracy and Error | 精度与误差
The error from truncating a binomial series is approximately the magnitude of the first omitted term. For an alternating series, this bound is particularly reliable.
截断二项级数所产生的误差大约等于被省略的第一项的大小。对于交错级数，这个误差界尤为可靠。
For example, in the expansion of (1 + x)1/2 .01, this term is about 10⁻⁶, so the approximation is accurate to around 6 decimal places.
例如，在 (1 + x)1/2 的展开式中，被省略的第一项通常是 x³ 量级。若 x = 0.01，该项约为 10⁻⁶，因此近似值精度大约到 6 位小数。
9. Common Mistakes | 常见错误
- Using the expansion when |x| ≥ 1: the series diverges and gives a completely wrong estimate.
- 在 |x| ≥ 1 时使用展开式：级数发散，给出的估算完全错误。
- Forgetting to factor out a value of a so that the bracket becomes (1 + x).
- 忘了提出 a 的因子，使得括号变成 (1 + x) 的形式。
- Writing incorrect binomial coefficients for fractional n, such as using ⁿC₂ instead of n(n−1)/2.
- 对分数 n 写出错误的二项式系数，例如写 ⁿC₂ 而不是 n(n−1)/2。
- Substituting the value of x incorrectly after a rearrangement.
- 在变形后错误地代入 x 的值。
10. Exam Tips | 考试技巧
In Edexcel A-Level exams, always state the range of validity when asked for a binomial expansion. Choose a convenient value of x that is small and satisfies the convergence condition.
在 Edexcel A-Level 考试中，当被要求进行二项展开时，务必注明有效范围。选择方便且满足收敛条件的小 x 值。
- Read the question carefully to see how many terms you need to keep.
- 仔细审题，弄清需要保留多少项。
- Use the exact form first, then substitute the decimal value at the final step.
- 先用精确形式展开，在最后一步再代入小数。
- Check your answer against a known value, e.g., √101 should be close to 10.
- 用已知值检查答案，例如 √101 应接近 10。
- Practise with past Edexcel questions to become fluent in rearranging expressions.
- 练习历年 Edexcel 真题，熟练掌握表达式变形。
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