Chapter 4 — Alternating Series, Absolute & Conditional Convergence

By the end of this chapter you will:

1. Apply the Alternating Series Test (AST)
2. Distinguish absolute vs. conditional convergence
3. Estimate the truncation error of an alternating series
4. Handle every AP question that combines AST with error bounds

4.1 What is an Alternating Series?

A series whose terms alternate sign:

$$\sum_{n=1}^{\infty} (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + \cdots \quad (b_n > 0)$$

or

$$\sum_{n=1}^{\infty} (-1)^{n} b_n = -b_1 + b_2 - b_3 + \cdots$$

Canonical example: $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ — the alternating harmonic series, which converges to $\ln 2$.

🎵 Audio 1 — What is an alternating series? — audio/ch4_01_intro.mp3

4.2 Alternating Series Test (AST)

Two hypotheses required:

1. $b_n$ is monotonically decreasing (i.e. $b_{n+1} \le b_n$ for sufficiently large $n$)
2. $\lim_{n\to\infty} b_n = 0$

Conclusion: $\sum (-1)^{n-1} b_n$ converges.

Intuition — pendulum: the partial sums oscillate around the limit with decreasing amplitude. As the swing shrinks to 0, the pendulum must settle at a single point.

🎵 Audio 2 — AST hypotheses + pendulum analogy — audio/ch4_02_ast.mp3

Example 1 — alternating harmonic

$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$.

• $b_n = \frac{1}{n}$ is monotonically decreasing ✓
• $\lim b_n = 0$ ✓

By AST, converges (sum $= \ln 2$).

Example 2

$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 1}$.

• $b_n = \frac{n}{n^2+1}$. Check monotonicity via $f(x) = \frac{x}{x^2+1}$, $f'(x) = \frac{1-x^2}{(x^2+1)^2} < 0$ for $x>1$. ✓
• $\lim b_n = 0$. ✓

Converges.

⚠️ Trap: many students forget to check monotonicity. If $b_n \to 0$ but is not eventually decreasing, AST does not apply.

🔑 Three ways to verify monotonicity: derivative sign, ratio $b_{n+1}/b_n$ vs. 1, or rewrite $b_n$ as a known monotone function.

4.3 Absolute vs. Conditional Convergence

For a series $\sum a_n$ (mixed signs allowed):

🎵 Audio 3 — Absolute vs. conditional — audio/ch4_03_abs_cond.mp3

Key relationship

$$\text{absolute convergence} \Rightarrow \text{convergence}$$

The converse is false — conditional convergence exists.

Comparison table

🔑 Workflow:

1. First test the absolute version $\sum \lvert a_n\rvert$ using Chapter 3 tools.
2. If it converges → absolutely convergent, done.
3. If it diverges → check $\sum a_n$ via AST. AST passes → conditional. AST fails → divergent.

🎵 Audio 4 — Three-step workflow — audio/ch4_04_workflow.mp3

4.4 Alternating Series Estimation Theorem (AST Error Bound)

Theorem: if $\sum (-1)^{n-1} b_n$ satisfies AST with sum $S$ and partial sum $S_N$, then

$$\lvert S - S_N\rvert \le b_{N+1}$$

In words: when you truncate an alternating series after $N$ terms, the truncation error is bounded by the first term you didn't add.

Intuition: the next pendulum swing is the maximum distance from the current position to the limit.

🎵 Audio 5 — AST error bound + worked example — audio/ch4_05_error_bound.mp3

Example (AP-style)

Estimate $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3}$ with the first 3 terms; bound the error.

Partial sum:
$$S_3 = 1 - \frac{1}{8} + \frac{1}{27} = \frac{197}{216} \approx 0.912$$

Error:
$$\lvert S - S_3 \rvert \le b_4 = \frac{1}{4^3} = \frac{1}{64} \approx 0.0156$$

🔑 Inverse problem: "How many terms do I need so the error is below $\varepsilon$?" → solve $b_{N+1} \le \varepsilon$.

Inverse example

How many terms of $\sum \frac{(-1)^{n-1}}{n^3}$ are needed so the partial-sum error is below $0.001$?

Solve $\frac{1}{(N+1)^3} < 0.001 \Rightarrow (N+1)^3 > 1000 \Rightarrow N+1 > 10 \Rightarrow N \ge 10$.

You need at least 10 terms.

4.5 Mixed Examples

Decide convergence; if convergent state absolute or conditional.

(a) $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$

(b) $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2 + 1}$

(c) $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n n}{n+1}$

Solutions:

(a) Absolute version $\sum \frac{1}{\sqrt{n}}$ is a $p=1/2$ series → diverges. AST: $b_n = 1/\sqrt{n}$ decreases to 0 → converges. Conditionally convergent.

(b) Absolute version $\sum \frac{1}{n^2+1}$ converges by comparison with $\frac{1}{n^2}$ → absolutely convergent.

(c) Term limit $\frac{(-1)^n n}{n+1}$ doesn't go to 0 (oscillates between $\pm 1$) → divergent by n-th term test.

🎵 Audio 6 — Walkthrough of (a)–(c) + recap — audio/ch4_06_examples_recap.mp3

4.6 Trap Alerts ⚠️

1. AST requires monotonicity. $b_n \to 0$ alone is not enough.
2. Test absolute convergence first. Skipping this on AP costs partial credit.
3. Error bound is $b_{N+1}$, the first omitted term — a common off-by-one trap.
4. Brackets: $S$ always lies between $S_{N}$ and $S_{N+1}$ (oscillating-pendulum visualization).
5. $(-1)^n$ alone diverges — don't even try AST.

4.7 Mnemonic

"Test the absolute first; AST needs monotonic decrease; error is the next term."

• Absolute first: handle $\sum \lvert a_n\rvert$ via Chapter 3 tools
• AST: monotonic decreasing and $\to 0$
• Error: $\lvert S - S_N\rvert \le b_{N+1}$

Media Inventory

End of chapter. Next: power series, radius and interval of convergence.