IB Mathematics: Key Points on Infinite Series | IB数学：无穷级数要点解析

📚 IB Mathematics: Key Points on Infinite Series | IB数学：无穷级数要点解析

Infinite series are a central topic in IB Mathematics Analysis and Approaches HL. Understanding how to determine convergence, evaluate sums, and apply series expansions is essential for exam success.

无穷级数是 IB 数学分析与方法（AA）HL 的核心内容。掌握如何判断收敛性、求和以及运用级数展开，是考试取得高分的关键。

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1. What Is an Infinite Series? | 什么是无穷级数？

An infinite series is the sum of the terms of an infinite sequence. Given a sequence (a_n), we define the series as (a₁ + a₂ + a₃ + cdots). The sequence of partial sums is (S_n = a₁ + a₂ + cdots + a_n).

无穷级数是无穷数列各项之和。给定数列 (a_n)，级数定义为 (a₁ + a₂ + a₃ + cdots)。部分和序列为 (S_n = a₁ + a₂ + cdots + a_n)。

If the partial sums (S_n) approach a finite limit (S) as (n to infty), the series converges and its sum is (S). Otherwise, it diverges.

若部分和 (S_n) 当 (n to infty) 时趋于有限极限 (S)，则级数收敛，其和为 (S)；否则级数发散。

(displaystyle S = lim_ntoinfty S_n = sum_n=1^infty a_n)

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2. The Divergence Test (n-th Term Test) | 发散检验（一般项检验）

The first test to apply is the divergence test: if (lim_ntoinfty a_n neq 0) (or the limit does not exist), then the series (sum a_n) diverges. However, if the limit is zero, the series may still converge or diverge; the test is inconclusive.

首先应使用发散检验：若 (lim_ntoinfty a_n neq 0)（或极限不存在），则级数 (sum a_n) 发散。但若极限为 0，级数仍可能收敛或发散，此检验无法判定。

For example, the series (sum frac{n}{n+1}) has terms approaching 1, so it diverges. But (sum frac{1}{n}) also has terms approaching 0, yet it diverges as well.

例如，级数 (sum frac{n}{n+1}) 的项趋于 1，因此发散。而 (sum frac{1}{n}) 的项虽趋于 0，但同样发散。

• If (a_n notto 0), the series must diverge.
• If (a_n to 0), further testing is required.

• 若 (a_n notto 0)，级数必发散。
• 若 (a_n to 0)，需进一步检验。

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3. Geometric Series | 几何级数

A geometric series has the form (sum_n=0^infty arⁿ). It converges if and only if (|r| < 1). Its sum is (frac{a}{1-r}). If (r geq 1) or (r leq -1), the series diverges.

几何级数的形式为 (sum_n=0^infty arⁿ)。它收敛当且仅当 (|r| < 1)，其和为 (frac{a}{1-r})。若 (r geq 1) 或 (r leq -1)，则级数发散。

(displaystyle sum_n=0^infty arⁿ = frac{a}{1-r}, quad |r|<1)

For example, (sum_n=0^infty left(frac{1}{2}right)ⁿ = 2). In IB questions, you may be asked to find the sum of a recurring decimal by expressing it as a geometric series.

例如，(sum_n=0^infty left(frac{1}{2}right)ⁿ = 2)。在 IB 考题中，常要求将循环小数表达为几何级数并求和。

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4. Harmonic Series and p-Series | 调和级数与 p 级数

The harmonic series (sum_n=1^infty frac{1}{n}) diverges, although its terms tend to zero. More generally, the p-series (sum_n=1^infty frac{1}{n^p}) converges if (p > 1) and diverges if (p leq 1).

调和级数 (sum_n=1^infty frac{1}{n}) 虽然项趋于 0，但却发散。更一般地，p 级数 (sum_n=1^infty frac{1}{n^p}) 当 (p > 1) 时收敛，当 (p leq 1) 时发散。

Series	Value of (p)	Convergence
(sum frac{1}{n})	(p=1)	Diverges
(sum frac{1}{n1.5})	(p=1.5)	Converges

级数	(p) 值	收敛性
(sum frac{1}{n})	(p=1)	发散
(sum frac{1}{n1.5})	(p=1.5)	收敛

This test is useful when comparing series with rational or radical terms.

该检验法在比较含有理式或根式的级数时非常有用。

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5. Comparison Test | 比较判别法

For series with non-negative terms, if (0 leq a_n leq b_n) for all sufficiently large (n), then:

对于非负项级数，若对所有足够大的 (n) 有 (0 leq a_n leq b_n)，则：

• If (sum b_n) converges, then (sum a_n) converges.
• If (sum a_n) diverges, then (sum b_n) diverges.

• 若 (sum b_n) 收敛，则 (sum a_n) 收敛。
• 若 (sum a_n) 发散，则 (sum b_n) 发散。

Example: (sum frac{1}{n²⁺¹}) converges because (frac{1}{n²⁺¹} leq frac{1}{n²}) and (sum frac{1}{n²}) converges.

例：(sum frac{1}{n²⁺¹}) 收敛，因为 (frac{1}{n²⁺¹} leq frac{1}{n²})，且 (sum frac{1}{n²}) 收敛。

In the limit comparison test, for positive (a_n, b_n), if (lim_ntoinfty frac{a_n}{b_n} = c) with (0 < c < infty), then both series behave the same.

在极限比较判别法中，对于正项 (a_n, b_n)，若 (lim_ntoinfty frac{a_n}{b_n} = c)，其中 (0 < c < infty)，则两个级数的敛散性相同。

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6. Ratio Test | 比值判别法

The ratio test is especially effective when terms involve factorials or exponential expressions. Let (L = lim_ntoinfty left|frac{a_n+1}{a_n}right|).

比值判别法特别适用于含阶乘或指数表达式的项。设 (L = lim_ntoinfty left|frac{a_n+1}{a_n}right|)。

If (L < 1), the series converges absolutely. If (L > 1), it diverges. If (L = 1), the test is inconclusive.

若 (L < 1)，级数绝对收敛；若 (L > 1)，级数发散；若 (L = 1)，检验失效。

For example, (sum_n=0^infty frac{2ⁿ}{n!}) converges because (left|frac{2ⁿ⁺¹/(n+1)!}{2ⁿ/n!}right| = frac{2}{n+1} to 0).

例如，(sum_n=0^infty frac{2ⁿ}{n!}) 收敛，因为 (left|frac{2ⁿ⁺¹/(n+1)!}{2ⁿ/n!}right| = frac{2}{n+1} to 0)。

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7. Integral Test | 积分判别法

If (f(x)) is positive, continuous, and decreasing for (x geq 1), and (a_n = f(n)), then the series (sum_n=1^infty a_n) converges if and only if the improper integral (int_1^infty f(x),dx) converges.

若 (f(x)) 在 (x geq 1) 上为正、连续且递减，且 (a_n = f(n))，则级数 (sum_n=1^infty a_n) 收敛当且仅当反常积分 (int_1^infty f(x),dx) 收敛。

For instance, take (f(x) = frac{1}{x^p}). The integral (int_1^infty x^-p dx) converges exactly when (p > 1), reinforcing the p-series rule.

例如，取 (f(x) = frac{1}{x^p})。积分 (int_1^infty x^-p dx) 恰在 (p > 1) 时收敛，这印证了 p 级数的规则。

This test is useful in IB when a series like (sum frac{1}{n ln n}) appears. Since (int_2^infty frac{dx}{x ln x}) diverges, the series diverges.

当遇到 (sum frac{1}{n ln n}) 这类级数时，积分判别法很有用。由于 (int_2^infty frac{dx}{x ln x}) 发散，因此该级数发散。

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8. Alternating Series | 交错级数

An alternating series has terms that alternate in sign, such as (sum_n=1^infty (-1)ⁿ⁺¹ a_n), where (a_n > 0). The alternating series test states that if (a_n) is decreasing and (lim_ntoinfty a_n = 0), then the series converges.

交错级数各项符号交替，例如 (sum_n=1^infty (-1)ⁿ⁺¹ a_n)，其中 (a_n > 0)。交错级数检验法指出：若 (a_n) 递减且 (lim_ntoinfty a_n = 0)，则级数收敛。

Example: (sum_n=1^infty frac{(-1)ⁿ⁺¹}{n}) converges to (ln 2). Without the alternating signs, the harmonic series diverges.

例：(sum_n=1^infty frac{(-1)ⁿ⁺¹}{n}) 收敛于 (ln 2)。若没有交替符号，调和级数则发散。

For partial sums of an alternating series, the error after (N) terms is at most the magnitude of the first omitted term (a_N+1). This is used in IB approximation questions.

对于交错级数的部分和，取前 (N) 项后的误差至多为第一项被舍去项 (a_N+1) 的大小。这在 IB 近似问题中经常用到。

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9. Absolute and Conditional Convergence | 绝对收敛与条件收敛

A series (sum a_n) converges absolutely if (sum |a_n|) converges. If (sum a_n) converges but (sum |a_n|) diverges, it converges conditionally.

若 (sum |a_n|) 收敛，则称级数 (sum a_n) 绝对收敛。若 (sum a_n) 收敛但 (sum |a_n|) 发散，则称其为条件收敛。

Every absolutely convergent series converges, but a conditionally convergent series may have rearrangements that converge to different sums. This is the Riemann rearrangement theorem.

每个绝对收敛的级数都收敛，但条件收敛的级数经过重新排列后可能收敛到不同的和，这就是黎曼重排定理。

• Absolute convergence: (sum frac{(-1)ⁿ}{n²}) converges absolutely.
• Conditional convergence: (sum frac{(-1)ⁿ}{n}) converges conditionally.

• 绝对收敛：(sum frac{(-1)ⁿ}{n²}) 绝对收敛。
• 条件收敛：(sum frac{(-1)ⁿ}{n}) 条件收敛。

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10. Power Series | 幂级数

A power series is of the form (sum_n=0^infty c_n (x-a)ⁿ). It converges for values of (x) within its radius of convergence (R), given by the ratio test or the root test.

幂级数的形式为 (sum_n=0^infty c_n (x-a)ⁿ)。它在收敛半径 (R) 内的 (x) 值处收敛，(R) 可由比值检验或根值检验求得。

(displaystyle R = frac{1}{limsup_ntoinfty sqrt[n]{|c_n|}})

The interval of convergence must be checked separately at the endpoints (x = a pm R).

收敛区间需单独检查端点 (x = a pm R) 处的敛散性。

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11. Taylor and Maclaurin Series | 泰勒级数与麦克劳林级数

If (f) has derivatives of all orders at (x=a), its Taylor series is:

若 (f) 在 (x=a) 处有任意阶导数，其泰勒级数为：

(displaystyle f(x) = sum_n=0^infty frac{f⁽ⁿ⁾(a)}{n!} (x-a)ⁿ)

When (a=0), the series is called a Maclaurin series. Common expansions include:

当 (a=0) 时，该级数称为麦克劳林级数。常见展开包括：

Function	Series	Radius of convergence
(ex)	(sumn=0infty frac{xn}{n!})	(infty)
(sin x)	(sumn=0infty (-1)n frac{x2n+1}{(2n+1)!})	(infty)
(cos x)	(sumn=0infty (-1)n frac{x2n}{(2n)!})	(infty)
(frac{1}{1-x})	(sumn=0infty xn)	(1)

函数	级数	收敛半径
(ex)	(sumn=0infty frac{xn}{n!})	(infty)
(sin x)	(sumn=0infty (-1)n frac{x2n+1}{(2n+1)!})	(infty)
(cos x)	(sumn=0infty (-1)n frac{x2n}{(2n)!})	(infty)
(frac{1}{1-x})	(sumn=0infty xn)	(1)

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12. Practical Exam Tips | 考试实用技巧

In IB exams, always identify the type of series before choosing a test. Write down the test you are using and verify its conditions explicitly.

在 IB 考试中，要先判断级数类型再选择检验方法。写出所用检验法，并明确验证其条件。

• Check the n-th term limit first: it can quickly rule out convergence.
• Use ratio test for factorials and powers like (n!) and (cⁿ).
• Use comparison or limit comparison for rational expressions.
• Remember endpoint checks for power series.
• Write the first few terms of a Maclaurin series to verify signs.

• 首先检查一般项极限：可快速排除收敛可能。
• 对含阶乘和幂如 (n!) 和 (cⁿ) 使用比值检验。
• 对有理表达式使用比较或极限比较。
• 幂级数需检查端点。
• 写出麦克劳林级数的前几项以验证符号。

Common mistakes include applying the divergence test backward, ignoring absolute values in the ratio test, and forgetting that an alternating harmonic series converges conditionally.

常见错误包括：反向使用发散检验、在比值检验中忽略绝对值，以及忘记交错调和级数是条件收敛的。

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