Binomial Expansion: Part 2 | 二项式展开：第二部分

📚 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。

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1. The General Binomial Theorem | 一般二项式定理

For a positive integer n, the binomial expansion is familiar: \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \dots + x^n\). 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: \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots\)

  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 \(x^r\) is written as \(\binom{n}{r} = \frac{n(n-1)(n-2)\dots(n-r+1)}{r!}\). 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。

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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 \(\sqrt{4+x}\) in the form \(2(1+\frac{x}{4})^{1/2}\), 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。

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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 \(T_r\)
(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 \(x^5\) is found by \(r=5\): \(C(8,5)(2^3)(-3)^5 = 56 \times 8 \times (-243) = -108864\).

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。

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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 \(\sqrt{1.04}\), write it as \((1+0.04)^{1/2} \approx 1 + \frac{1}{2}(0.04) + \frac{(1/2)(-1/2)}{2}(0.04)^2 = 1 + 0.02 – 0.0002 = 1.0198\).

  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。

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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 \(\frac{2x+3}{(1-x)(1+2x)}\).

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。

\(\frac{2x+3}{(1-x)(1+2x)} = \frac{A}{1-x} + \frac{B}{1+2x}\)

Solving gives \(A = \frac{5}{3}\), \(B = \frac{1}{3}\). 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 \(\frac{A}{1-x}\), we use \((1-x)^{-1} = 1 + x + x^2 + x^3 + \dots\), 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 \(\frac{B}{1+2x}\), rewrite as \(B(1+2x)^{-1}\), expanding to get \(B(1 – 2x + 4x^2 – 8x^3 + \dots)\), valid for \(|2x| < 1\), i.e. \(|x| < \frac{1}{2}\).

  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。

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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: \(\binom{n}{r} = \binom{n}{n-r}\). This holds for integers \(0 \le r \le n\).

  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: \(\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}\).

  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 \(2^n\): \(\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n\).

  (1+x)ⁿ 中所有系数之和为 2ⁿ：C(n,0) + C(n,1) + … + C(n,n) = 2ⁿ。

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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 \(1 + 2x + 3x^2 + 4x^3 + \dots\) 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}。

\(\sum_{k=0}^{\infty} (k+1)x^k = \frac{1}{(1-x)^2}\)

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。

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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。

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9. Worked Example: Expanding \(\sqrt{4+x}\) | 例题精讲：展开 √(4+x)

Find the first three terms in the expansion of \(\sqrt{4+x}\) and state the range of validity.

求 √(4+x) 展开式的前三项，并说明有效范围。

Rewrite: \(\sqrt{4+x} = 2\sqrt{1+\frac{x}{4}} = 2(1+\frac{x}{4})^{1/2}\).

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：

\(2\left[1 + \frac{1}{2}\left(\frac{x}{4}\right) + \frac{(\frac{1}{2})(-\frac{1}{2})}{2}\left(\frac{x}{4}\right)^2 + \dots\right]\)

This simplifies to \(2 + \frac{x}{4} – \frac{x^2}{64} + \dots\). 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。

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10. Worked Example: Approximation of a Root | 例题精讲：根的近似计算

Use the binomial expansion to approximate \(\sqrt[3]{1.02}\) to four decimal places.

使用二项式展开将 ∛1.02 近似到四位小数。

Let \(\sqrt[3]{1.02} = (1+0.02)^{1/3}\). The expansion is:

设 ∛1.02 = (1+0.02)^{1/3}。展开式为：

\(1 + \frac{1}{3}(0.02) + \frac{(\frac{1}{3})(-\frac{2}{3})}{2}(0.02)^2 + \frac{(\frac{1}{3})(-\frac{2}{3})(-\frac{5}{3})}{6}(0.02)^3\)

Calculating: \(1 + 0.0066667 – 0.0000444 + 0.0000007 = 1.006623\). Hence \(\sqrt[3]{1.02} \approx 1.0066\) 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）。

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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 \(\frac{1}{1+2x}\) up to the term in \(x^3\), 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 \(x^4\) in the expansion of \((1-3x)^{-\frac{1}{2}}\).

  求 (1-3x)^{-1/2} 展开式中 x⁴ 的系数。

• Use partial fractions to expand \(\frac{1}{(2-x)(1+x)}\) up to the term in \(x^2\).

  使用部分分式将 1/[(2-x)(1+x)] 展开至 x² 项。

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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。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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