Binomial Estimation | 二项式估计

📚 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

This familiar form is the foundation for the general binomial series. 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), the first omitted term is typically of order x³. If x = 0.01, 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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Binomial Estimation | 二项式估计