📚 Taylor Series and Function Approximation | IB数学：泰勒级数与函数近似
Many mathematical functions cannot be evaluated exactly by simple arithmetic. Taylor series provide a systematic way to approximate any sufficiently smooth function using polynomials, which are far easier to compute, differentiate, and integrate.
许多数学函数无法通过简单算术精确求值。泰勒级数提供了一种系统方法，用容易计算、求导和积分的多项式来逼近任何足够光滑的函数。
1. What Is a Taylor Series? | 什么是泰勒级数
For a function f that is infinitely differentiable at a point a, its Taylor series is written as
f(x) = Σₙ₌₀᪾ f⁽ⁿ⁾(a)/n! × (x − a)ⁿ
where f⁽ⁿ⁾(a) denotes the n-th derivative evaluated at a, and n! is the factorial of n. This infinite sum represents the function exactly on its interval of convergence.
对于在点 a 处无穷可微的函数 f，其泰勒级数写为上面的形式。其中 f⁽ⁿ⁾(a) 表示在 a 处求值的第 n 阶导数，n! 是 n 的阶乘。该无穷和在其收敛区间上精确表示原函数。
2. Maclaurin Series: The Special Case a = 0 | 麦克劳林级数：a = 0 的特殊情形
When the expansion point is chosen as a = 0, the Taylor series becomes the Maclaurin series:
f(x) = Σₙ₌₀᪾ f⁽ⁿ⁾(0)/n! × xⁿ
Common Maclaurin expansions appear frequently in IB examinations. For example, eˣ = Σₙ₌₀᪾ xⁿ/n!, which converges for all real numbers.
当展开点取 a = 0 时，泰勒级数即为麦克劳林级数。常见的麦克劳林展开在 IB 考试中频繁出现。例如 eˣ = Σₙ₌₀᪾ xⁿ/n!，它对所有实数收敛。
3. Geometric Intuition of Taylor Polynomials | 泰勒多项式的几何直觉
The first-order Taylor polynomial is the tangent line at a; it matches the function’s value and its first derivative. The second-order polynomial adds curvature by matching f″(a), giving a closer fit near a.
一阶泰勒多项式是函数在 a 处的切线；它匹配函数值和一阶导数。二阶多项式通过匹配 f″(a) 来加入曲率信息，使函数在 a 附近拟合得更紧密。
Each additional term corrects the approximation for one more order of behaviour. As n increases, the approximation becomes more accurate near the centre, though it may still diverge far away.
每增加一项，就修正一个更高阶的行为。随着 n 增大，近似在中心附近越发精确，但在远离中心处仍可能发散。
4. A Concrete Example: Approximating eˣ | 具体例子：逼近 eˣ
Consider f(x) = eˣ and expand about a = 0. The first three non-zero terms give
eˣ ≈ 1 + x + x²/2 + x³/6
At x = 0.1, this polynomial yields 1 + 0.1 + 0.005 + 0.0001666667 = 1.10516667, while the true value is 1.10517092. The error is only about 4.2 × 10⁻⁶.
取 f(x) = eˣ，在 a = 0 处展开。前三项给出上面的多项式近似。在 x = 0.1 处，计算结果为 1.10516667，真实值为 1.10517092，误差仅为约 4.2 × 10⁻⁶。
Notice how rapidly the approximation improves. Including the x⁴/24 term raises the estimate to 1.10517083, reducing the error by a further factor of ten.
注意近似改善的速度之快。加入 x⁴/24 项后估计值升至 1.10517083，误差又缩小了一个数量级。
5. Convergence: When Does a Taylor Series Work? | 收敛性：泰勒级数何时有效
A Taylor series may converge for all x (as with eˣ, sin x, cos x), for a finite interval (as with ln(1+x) on −1 < x ≤ 1), or only at the centre itself. The radius of convergence R is found using the ratio test:
R = limₙ→∞ |aₙ / aₙ₊₁|
where aₙ = f⁽ⁿ⁾(a)/n!. For series like ln(1+x), the radius is 1, but the endpoint x = 1 converges conditionally while x = −1 diverges.
泰勒级数可能对所有 x 收敛（如 eˣ、sin x、cos x），可能在有限区间内收敛（如 ln(1+x) 在 −1 < x ≤ 1），也可能仅在中心点收敛。收敛半径 R 可通过比值检验求得。例如 ln(1+x) 的收敛半径为 1，端点 x = 1 条件收敛，而 x = −1 发散。
6. Lagrange Error Bound: How Accurate Is the Approximation? | 拉格朗日余项：近似有多精确
The difference between a function and its n-th Taylor polynomial is the remainder Rₙ(x), given by the Lagrange form:
Rₙ(x) = f⁽ⁿ⁺¹⁾(c)/(n+1)! × (x − a)ⁿ⁺¹
where c is some number strictly between a and x. In IB exams, you are often asked to find the maximum possible error on a given interval using this formula.
函数与其 n 阶泰勒多项式之差称为余项 Rₙ(x)，其拉格朗日形式为上式，其中 c 是介于 a 和 x 之间的某个数。在 IB 考试中，常要求利用该公式求给定区间上的最大可能误差。
Key strategy: bound |f⁽ⁿ⁺¹⁾(c)| by its maximum value on the relevant interval, then compute the resulting upper bound for Rₙ(x). This gives a rigorous error estimate, not just a heuristic one.
关键策略：用 |f⁽ⁿ⁺¹⁾(c)| 在相应区间上的最大值来界定，然后计算 Rₙ(x) 的上界。这给出的是严格的误差估计，而非仅仅经验性判断。
7. Euler’s Formula: Taylor Series in Complex Analysis | 欧拉公式：复数分析中的泰勒级数
By substituting ix into the Maclaurin series for eˣ and separating real and imaginary parts, one obtains one of the most elegant results in mathematics:
eⁱˣ = cos x + i sin x
Setting x = π yields eⁱᵖ + 1 = 0, the famous identity linking the five fundamental constants. This derivation relies entirely on series rearrangement, a technique IB students should be comfortable with.
将 ix 代入 eˣ 的麦克劳林级数，分离实部与虚部，得到数学中最优雅的结果之一：eⁱˣ = cos x + i sin x。令 x = π 得 eⁱᵖ + 1 = 0，这是联系五个基本常数的著名恒等式。该推导完全依赖级数重排，IB 学生应熟练掌握这一技巧。
8. Numerical Methods: Linearisation and Euler’s Method | 数值方法：线性化与欧拉法
In differential equations, the first-order Taylor approximation f(x) ≈ f(a) + f′(a)(x − a) is exactly the tangent-line linearisation used to estimate solution curves locally.
在微分方程中，一阶泰勒近似 f(x) ≈ f(a) + f′(a)(x − a) 正是用于局部估计解曲线的切线线性化。
Euler’s method for solving y′ = f(x, y) iterates this idea: each step advances the solution along its tangent line. Higher-order Taylor methods, which use additional derivative terms, improve accuracy but require more computation.
欧拉法求解微分方程 y′ = f(x, y) 正是迭代该思想：每一步沿切线推进。更高阶的泰勒方法使用更多导数项提高精度，但计算量更大。
This connection from pure series to applied numerical analysis reflects the interdisciplinary nature of the IB Mathematics AA curriculum.
从纯级数到应用数值分析的这一联系，体现了 IB 数学 AA 课程跨学科的特点。
9. Special Expansions: ln(1+x) and arctan x | 特殊展开：ln(1+x) 与 arctan x
Two expansions worth memorising are
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + …
arctan x = x − x³/3 + x⁵/5 − x⁷/7 + …
Both converge only on |x| ≤ 1 (for arctan, all points; for ln, including x = 1 but excluding x = −1). These series are ideal for estimating π: substituting x = 1 into arctan gives π/4 = 1 − 1/3 + 1/5 − 1/7 + …
两个值得记忆的展开是 ln(1+x) 和 arctan x 的麦克劳林级数。二者仅在 |x| ≤ 1 上收敛（arctan 在所有端点收敛；ln 包含 x = 1 但排除 x = −1）。这些级数特别适合估计 π：将 x = 1 代入 arctan 的级数，得到 π/4 = 1 − 1/3 + 1/5 − 1/7 + …
10. Comparing Approximation Quality: A Table | 近似质量对比：一个表格
The table below compares the n-th Taylor polynomial of eˣ at x = 0.5 with the exact value.
下表比较 eˣ 在 x = 0.5 处的 n 阶泰勒多项式与精确值。
| n | Tₙ(0.5) | Absolute error | 绝对误差 |
| 0 | 1 | 0.64872 | 0.64872 |
| 1 | 1.5 | 0.14872 | 0.14872 |
| 2 | 1.625 | 0.02372 | 0.02372 |
| 3 | 1.645833 | 0.002887 | 0.002887 |
| 4 | 1.648438 | 0.000282 | 0.000282 |
The error decreases by roughly a factor of 1/(n+1) each step near x = 0.5, demonstrating the rapid improvement.
在 x = 0.5 附近，每一步误差约缩小到原来的 1/(n+1)，这体现了近似的快速改善。
11. IB Exam Strategy: Common Pitfalls | IB 考试策略：常见失分点
Students frequently forget the factorial denominators, apply the ratio test with the wrong limit, or disregard the interval of convergence when evaluating a series at a boundary point. Another common error is confusing Taylor series with Maclaurin series when the centre is not zero.
学生常忘记阶乘分母、在比值检验中用错极限，或在端点处求值时忽略收敛区间。另一个常见错误是当中心点不为零时，混淆泰勒级数与麦克劳林级数。
To avoid slips: first write the nth derivative pattern explicitly, check the centre, and always state the interval of convergence before substituting numeric values.
为避免失误：先明确写出第 n 阶导数的规律，检查中心点，并在代入数值前始终说明收敛区间。
12. Summary | 小结
Taylor series transform complicated functions into polynomials that can be evaluated, differentiated, and integrated easily. The key concepts — the expansion formula, Lagrange error bound, and convergence interval — appear in both Paper 2 and Paper 3 of IB Mathematics AA HL.
泰勒级数把复杂函数转化为容易求值、求导和积分的多项式。核心概念——展开公式、拉格朗日误差界和收敛区间——在 IB 数学 AA HL 的 Paper 2 和 Paper 3 中都会出现。
Mastering these ideas not only prepares you for examinations but also builds a foundation for university-level analysis and numerical mathematics.
掌握这些思想不仅为考试做好准备，也为大学阶段的分析学和数值数学打基础。
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