An alternating series is a series whose terms alternate in sign: $b_1 - b_2 + b_3 - b_4 + \cdots$ or $-b_1 + b_2 - b_3 + \cdots$. The Alternating Series Test provides a simple way to determine convergence for such series, and the Alternating Series Error Bound tells us how close a partial sum is to the total sum.
🎯 In this section you will learn
- The form of an alternating series: $\sum (-1)^{n-1} b_n$ with $b_n > 0$
- The Alternating Series Test (Leibniz Test)
- The Alternating Series Error Bound: $|R_n| \le b_{n+1}$
- How to estimate the sum of an alternating series
📌 Alternating Series Test (Leibniz Test)
$$ \text{For } \sum_{n=1}^\infty (-1)^{n-1} b_n \text{ with } b_n > 0, $$
$$ \text{if } b_{n+1} \le b_n \text{ for all } n \text{ and } \lim_{n \to \infty} b_n = 0, $$
$$ \text{then the series converges.} $$
💡 The Two Conditions
- The terms $b_n$ must be decreasing ($b_{n+1} \le b_n$)
- The terms must approach zero ($\lim b_n = 0$)
Test $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ for convergence.
①
Check decreasing
$b_n = \frac{1}{n}$, $b_{n+1} = \frac{1}{n+1} \le \frac{1}{n}$ ✓
②
Check limit
$\lim_{n \to \infty} \frac{1}{n} = 0$ ✓
③
Conclusion
The alternating harmonic series converges. (The regular harmonic series diverges!)
Test $\sum_{n=1}^\infty \frac{(-1)^{n-1} n}{n+1}$ for convergence.
- $b_n = \frac{n}{n+1}$, $\lim b_n = 1 \neq 0$
- Since the second condition fails, the series diverges (by nth term test).
📌 Alternating Series Error Bound
$$ \text{If } S = \sum_{n=1}^\infty (-1)^{n-1} b_n \text{ converges by AST, then } |S - S_N| \le b_{N+1} $$
The error after $N$ terms is less than the first omitted term
💡 Intuition
The partial sums of an alternating series "staircase" toward the total sum, and each new term is larger than the remaining error.
Approximate $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}$ using $S_4 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} = 0.58333$. Find the error bound.
- First omitted term: $b_5 = \frac{1}{5} = 0.2$
- Error bound: $|S - S_4| \le 0.2$
- The actual sum is $\ln 2 \approx 0.6931$, and $|0.6931 - 0.5833| = 0.1098 \le 0.2$ ✓
For $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2}$, how many terms are needed to approximate the sum with error $< 0.01$?
①
Error bound
$|R_N| \le b_{N+1} = \frac{1}{(N+1)^2}$
②
Set inequality
$\frac{1}{(N+1)^2} < 0.01$ → $(N+1)^2 > 100$ → $N+1 > 10$ → $N > 9$
③
Conclusion
$N = 10$ terms guarantee error less than $0.01$.
⚠️ Common Mistakes
- Forgetting to check decreasing condition: Even if $b_n \to 0$, the series may diverge if not decreasing (e.g., $b_n = 1/n$ for odd $n$, $b_n = 0$ for even $n$? Actually need to check).
- Confusing alternating series with absolute convergence: An alternating series may converge conditionally (like the alternating harmonic series).
- Error bound: $|R_N| \le b_{N+1}$ only applies when the Alternating Series Test conditions are satisfied.
🔍 Key Takeaways
- Alternating Series Test: If $b_n$ decreases to $0$, then $\sum (-1)^{n-1} b_n$ converges.
- Error bound: $|R_N| \le b_{N+1}$ — the first omitted term.
- Alternating series often converge conditionally (not absolutely).
- This test is simple and powerful — memorize it!
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