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。

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 \(a_1 + a_2 + a_3 + \cdots\). The sequence of partial sums is \(S_n = a_1 + a_2 + \cdots + a_n\).

无穷级数是无穷数列各项之和。给定数列 \(a_n\)，级数定义为 \(a_1 + a_2 + a_3 + \cdots\)。部分和序列为 \(S_n = a_1 + a_2 + \cdots + a_n\)。

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。

\(\displaystyle S = \lim_{n\to\infty} S_n = \sum_{n=1}^{\infty} a_n\)

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 \(\lim_{n\to\infty} a_n \neq 0\) (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 \(\lim_{n\to\infty} a_n \neq 0\)（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 \(a_n \not\to 0\), the series must diverge.
If \(a_n \to 0\), further testing is required.

若 \(a_n \not\to 0\)，级数必发散。
若 \(a_n \to 0\)，需进一步检验。

3. Geometric Series | 几何级数

A geometric series has the form \(\sum_{n=0}^{\infty} ar^n\). 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 \(\sum_{n=0}^{\infty} ar^n\)。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。

\(\displaystyle \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r|<1\)

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。

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
\(\sum \frac{1}{n^{1.5}}\)	\(p=1.5\)	Converges

级数	\(p\) 值	收敛性
\(\sum \frac{1}{n}\)	\(p=1\)	发散
\(\sum \frac{1}{n^{1.5}}\)	\(p=1.5\)	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。

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: \(\sum \frac{1}{n^2+1}\) converges because \(\frac{1}{n^2+1} \leq \frac{1}{n^2}\) and \(\sum \frac{1}{n^2}\) converges.

例：\(\sum \frac{1}{n^2+1}\) 收敛，因为 \(\frac{1}{n^2+1} \leq \frac{1}{n^2}\)，且 \(\sum \frac{1}{n^2}\) 收敛。

In the limit comparison test, for positive \(a_n, b_n\), if \(\lim_{n\to\infty} \frac{a_n}{b_n} = c\) 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 \(\lim_{n\to\infty} \frac{a_n}{b_n} = c\)，其中 \(0 < c < \infty\)，则两个级数的敛散性相同。

6. Ratio Test | 比值判别法

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

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

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

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

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。

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 \(\lim_{n\to\infty} a_n = 0\), 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 \(\lim_{n\to\infty} a_n = 0\)，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。

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: \(\sum \frac{(-1)^n}{n^2}\) converges absolutely.
Conditional convergence: \(\sum \frac{(-1)^n}{n}\) converges conditionally.

绝对收敛：\(\sum \frac{(-1)^n}{n^2}\) 绝对收敛。
条件收敛：\(\sum \frac{(-1)^n}{n}\) 条件收敛。

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。

\(\displaystyle R = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|c_n|}}\)

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。

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
\(e^x\)	\(\sum_{n=0}^{\infty} \frac{x^n}{n!}\)	\(\infty\)
\(\sin x\)	\(\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\)	\(\infty\)
\(\cos x\)	\(\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\)	\(\infty\)
\(\frac{1}{1-x}\)	\(\sum_{n=0}^{\infty} x^n\)	\(1\)

函数	级数	收敛半径
\(e^x\)	\(\sum_{n=0}^{\infty} \frac{x^n}{n!}\)	\(\infty\)
\(\sin x\)	\(\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\)	\(\infty\)
\(\cos x\)	\(\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\)	\(\infty\)
\(\frac{1}{1-x}\)	\(\sum_{n=0}^{\infty} x^n\)	\(1\)

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 \(c^n\).
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 \(c^n\) 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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IB Mathematics: Key Points on Infinite Series | IB数学：无穷级数要点解析