📚 Binomial Expansion: Part 2 | 二项式展开：第二部分
The binomial theorem is a cornerstone of algebra, and its advanced applications often appear in A-Level mathematics examinations. In this article, we move beyond simple integer powers to explore the general binomial expansion, its validity, and its powerful use in approximations and series.
二项式定理是代数的基石，其高级应用在 A-Level 数学考试中经常出现。在本文中，我们将超越简单的整数次幂，探讨一般二项式展开、其适用范围以及在近似计算和级数中的强大用途。
1. The General Binomial Theorem | 一般二项式定理
For a positive integer n, the binomial expansion is familiar: (1+x)n = 1 + nx + n(n-1)2!x2 + … + xn. However, the theorem can be extended to any real exponent n using the generalised binomial coefficient.
对于正整数 n，二项式展开是熟悉的：(1+x)ⁿ = 1 + nx + n(n-1)/2!·x² + … + xⁿ。然而，该定理可以扩展到任意实数指数 n，只需使用广义二项式系数。
For any real n and |x| < 1, the expansion is: (1+x)n = 1 + nx + n(n-1)2!x2 + n(n-1)(n-2)3!x3 + …
对于任意实数 n 且 |x| < 1，展开式为：(1+x)ⁿ = 1 + nx + n(n-1)/2!·x² + n(n-1)(n-2)/3!·x³ + …
The coefficient of xr is written as binomnr = n(n-1)(n-2)…(n-r+1)r!. When n is not a positive integer, this series does not terminate.
xʳ 的系数写作 C(n,r) = n(n-1)(n-2)…(n-r+1)/r!。当 n 不是正整数时，该级数不会终止。
2. Validity of the Expansion | 展开的适用范围
Unlike the finite expansion for positive integer powers, the general binomial series is an infinite series, and it converges only under certain conditions.
与正整数次幂的有限展开不同，一般二项式级数是无穷级数，并且只在特定条件下收敛。
For integer n ≥ 0, the expansion is valid for all x because it terminates as a polynomial.
对于整数 n ≥ 0，展开式对所有 x 都有效，因为它是一个多项式，项数有限。
For non-integer n, the series converges when |x| < 1. At |x| = 1, convergence depends on the value of n.
对于非整数 n，级数在 |x| < 1 时收敛。在 |x| = 1 时，收敛性取决于 n 的值。
When rewriting an expression like √4+x in the form 2(1+x4)1/2, we must ensure that the series is used only where |x/4| < 1, i.e. |x| < 4.
当将 √4+x 改写为 2(1+x/4)1/2 的形式时，必须确保级数仅在 |x/4| < 1，即 |x| < 4 时使用。
3. Finding Specific Terms | 求特定项
In many problems, you are not asked for the whole expansion but for the coefficient of a particular power of x. The general term formula is your best tool.
在许多问题中，并非要求写出整个展开式，而是要求 x 的特定次幂的系数。通项公式是最佳工具。
| Expression | General term Tr |
| (a+b)ⁿ | C(n,r)aⁿ⁻ʳ bʳ |
| (1+kx)ⁿ | C(n,r) kʳ xʳ |
For example, in the expansion of (2-3x)8, the coefficient of x5 is found by r=5: C(8,5)(23)(-3)5 = 56 × 8 × (-243) = -108864.
例如，在 (2-3x)⁸ 的展开式中，x⁵ 的系数可由 r=5 得到：C(8,5)·2³·(-3)⁵ = 56 × 8 × (-243) = -108864。
4. Approximations Using the Binomial Expansion | 利用二项式展开求近似值
The binomial series is extremely useful for approximating roots and powers near 1. By taking the first few terms, we obtain a polynomial that is easy to compute.
二项式级数对于近似计算接近 1 的根和幂极为有用。取前几项即可得到一个容易计算的多项式。
To approximate √1.04, write it as (1+0.04)1/2 ≈ 1 + 12(0.04) + (1/2)(-1/2)2(0.04)2 = 1 + 0.02 - 0.0002 = 1.0198.
要近似计算 √1.04，可写成 (1+0.04)1/2 ≈ 1 + ½(0.04) + (½)(-½)/2·(0.04)² = 1 + 0.02 - 0.0002 = 1.0198。
The more terms used, the better the approximation, provided the series converges. For |x| < 1, the terms generally decrease in size.
使用越多项，近似效果越好，前提是级数收敛。对于 |x| < 1，各项通常逐渐减小。
5. Combining with Partial Fractions | 与部分分式结合
A rational function can often be decomposed into partial fractions, each of which can then be expanded binomially. This allows expansions of expressions like 2x+3(1-x)(1+2x).
一个有理函数通常可以分解为部分分式，然后每个分式可以用二项式定理展开。这使我们能展开如 (2x+3)/[(1-x)(1+2x)] 这样的表达式。
2x+3(1-x)(1+2x) = A1-x + B1+2x
Solving gives A = 53, B = 13. Then each fraction is expanded separately with appropriate validity ranges.
解出 A = 5/3，B = 1/3。然后每个分式分别展开，并注意各自有效的范围。
For A1-x, we use (1-x)-1 = 1 + x + x2 + x3 + …, valid for |x| < 1.
对于 A/(1-x)，使用 (1-x)-1 = 1 + x + x² + x³ + …，有效范围 |x| < 1。
For B1+2x, rewrite as B(1+2x)-1, expanding to get B(1 - 2x + 4x2 - 8x3 + …), valid for |2x| < 1, i.e. |x| < 12.
对于 B/(1+2x)，改写为 B(1+2x)-1，展开得到 B(1 - 2x + 4x² - 8x³ + …)，有效范围 |2x| < 1，即 |x| < 1/2。
6. Binomial Coefficients and Their Properties | 二项式系数及其性质
Binomial coefficients satisfy many identities that are frequently tested in examination problems.
二项式系数满足许多恒等式，这些恒等式在考试问题中经常被考查。
The symmetry property: binomnr = binomnn-r. This holds for integers 0 ≤ r ≤ n.
对称性质：C(n,r) = C(n,n-r)。该性质对所有整数 0 ≤ r ≤ n 成立。
Pascal's identity: binomnr + binomnr+1 = binomn+1r+1.
帕斯卡恒等式：C(n,r) + C(n,r+1) = C(n+1,r+1)。
Sum of all coefficients in (1+x)n is 2n: binomn0 + binomn1 + … + binomnn = 2n.
(1+x)ⁿ 中所有系数之和为 2ⁿ：C(n,0) + C(n,1) + … + C(n,n) = 2ⁿ。
7. Summation of Series | 级数求和
The binomial expansion can be used to sum certain infinite series by recognising the pattern of coefficients.
通过识别系数的模式，二项式展开可用于计算某些无穷级数的和。
For example, the series 1 + 2x + 3x2 + 4x3 + … is the derivative of (1-x)-1. Hence it equals (1-x)-2 for |x| < 1.
例如，级数 1 + 2x + 3x² + 4x³ + … 是 (1-x)-1 的导数。因此，对于 |x| < 1，它等于 (1-x)-2。
∑k=0∞ (k+1)xk = 1⁄(1-x)2
Similarly, multiplying (1+x)-1 by itself gives the series with alternating signs.
类似地，(1+x)-1 自乘得到具有交替符号的级数。
8. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Students often lose marks by forgetting the validity condition or misapplying the binomial coefficient when n is negative or fractional.
学生在 n 为负数或分数时，常因忘记有效性条件或错误应用二项式系数而失分。
Always state |x| < 1 when using the general binomial expansion, and adjust it for expressions like (a+bx)n.
使用一般二项式展开时，务必说明 |x| < 1，并对形如 (a+bx)ⁿ 的表达式作相应调整。
When n is not a positive integer, the expansion is infinite; never treat it as a polynomial with a last term.
当 n 不是正整数时，展开是无穷的；切勿将其视为有末项的多项式。
Check the sign: in (1-x)n, the signs alternate because each factor contributes a negative x.
检查符号：在 (1-x)ⁿ 中，由于每个因子都贡献一个负的 x，符号会交替变化。
9. Worked Example: Expanding √4+x | 例题精讲：展开 √4+x
Find the first three terms in the expansion of √4+x and state the range of validity.
求 √4+x 展开式的前三项，并说明有效范围。
Rewrite: √4+x = 2√1+x4 = 2(1+x4)1/2.
改写：√4+x = 2√1+x/4 = 2(1+x/4)1/2。
Using the general binomial theorem:
使用一般二项式定理：
2[1 + 12(x4) + (12)(-12)⁄2(x4)2 + …]
This simplifies to 2 + x4 - x2⁄64 + …. The series is valid when |x/4| < 1, i.e. |x| < 4.
这简化为 2 + x/4 - x²/64 + …。级数在 |x/4| < 1 即 |x| < 4 时有效。
10. Worked Example: Approximation of a Root | 例题精讲：根的近似计算
Use the binomial expansion to approximate 3√1.02 to four decimal places.
使用二项式展开将 ∛1.02 近似到四位小数。
Let 3√1.02 = (1+0.02)1/3. The expansion is:
设 ∛1.02 = (1+0.02)1/3。展开式为：
1 + 13(0.02) + (13)(-23)⁄2(0.02)2 + (13)(-23)(-53)⁄6(0.02)3
Calculating: 1 + 0.0066667 - 0.0000444 + 0.0000007 = 1.006623. Hence 3√1.02 ≈ 1.0066 to 4 d.p.
计算：1 + 0.0066667 - 0.0000444 + 0.0000007 = 1.006623。因此 ∛1.02 ≈ 1.0066（四舍五入到四个小数位）。
11. Practice Problems | 练习题目
Test your understanding with these questions.
用以下问题检验你的理解。
Expand 11+2x up to the term in x3, and state the valid range for x.
将 1/(1+2x) 展开至 x³ 项，并说明 x 的有效范围。
Find the coefficient of x4 in the expansion of (1-3x)-12.
求 (1-3x)-1/2 展开式中 x⁴ 的系数。
Use partial fractions to expand 1(2-x)(1+x) up to the term in x2.
使用部分分式将 1/[(2-x)(1+x)] 展开至 x² 项。
12. Conclusion | 总结
Mastering the general binomial expansion opens the door to solving a wide range of problems involving approximations, rational functions, and series. Remember to check the validity range and to handle fractional and negative exponents with care.
掌握一般二项式展开，为解决涉及近似、有理函数和级数的各类问题打开了大门。切记检查有效性范围，并谨慎处理分数和负指数。
With regular practice, the binomial theorem becomes one of the most reliable tools in your A-Level mathematics toolkit.
通过定期练习，二项式定理将成为你 A-Level 数学工具箱中最可靠的工具之一。
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