Alternating Series Test
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 5.5: “Alternating Series” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/5-5-alternating-series |
| Textbook used in class | Stewart, Calculus, Section 11.5: “Alternating Series” (Examples 1, 2, 3) |
Opening Scenario
The harmonic series $1 + 1/2 + 1/3 + \cdots$ diverges: the partial sums grow without bound. But the alternating harmonic series $1 - 1/2 + 1/3 - 1/4 + \cdots$ converges to $\ln 2$. The signs alternate, and successive partial sums overshoot and undershoot the limit by ever-smaller amounts, squeezing the series to a definite value. Alternating series can converge even when the positive version diverges.
Quick Reference
Alternating Series Test (Leibniz). A series of the form \[ \sum_{n=1}^{\infty} (-1)^{n-1} b_n = b_1 - b_2 + b_3 - b_4 + \cdots, \quad b_n > 0, \] converges if both conditions hold:
- $b_{n+1} \leq b_n$ for all $n$ (the terms decrease).
- $\displaystyle\lim_{n\to\infty} b_n = 0$.
Alternating Series Estimation Theorem. If the series satisfies the above conditions, the error made by stopping at the $n$-th partial sum satisfies \[ |R_n| = |s - s_n| \leq b_{n+1}. \] The error is at most the absolute value of the first omitted term, and the true sum lies between consecutive partial sums $s_n$ and $s_{n+1}$.
Key Concepts
1. Why Alternating Signs Help
The even partial sums $s_{2n}$ form an increasing sequence bounded above by $b_1$. The odd partial sums $s_{2n-1}$ form a decreasing sequence bounded below by $0$. Both sequences converge to the same limit because $s_{2n+1} - s_{2n} = b_{2n+1} \to 0$. The limit is sandwiched between every consecutive pair of partial sums.
2. Checking the Two Conditions
Both conditions are necessary. If $b_n \not\to 0$, the series diverges by the Divergence Test, regardless of the alternating signs. If the terms do not decrease, the argument above breaks down.
Example 1. Does $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$ converge? (Stewart 11.5, Example 1.)
Check: $b_n = 1/n$. Clearly $b_{n+1} = 1/(n+1) < 1/n = b_n$ (decreasing), and $b_n \to 0$.
Both conditions hold, so the alternating harmonic series converges.
Boxed answer: Converges (to $\ln 2$).
Example 2. Does $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n \cdot 3n}{4n - 1}$ converge? (Stewart 11.5, Example 2.)
Check condition 2. $b_n = 3n/(4n-1) \to 3/4 \neq 0$.
The terms do not tend to zero, so the series diverges by the Divergence Test.
Boxed answer: Diverges.
Example 3. Estimate $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^5}$ to within error $0.001$. (Stewart 11.5, Example 3.)
Goal. Find the first $n$ such that $b_{n+1} = 1/(n+1)^5 \leq 0.001$.
At $n = 3$: $b_4 = 1/4^5 = 1/1024 \approx 0.00098 < 0.001$.
So using three terms suffices: \[ s_3 = 1 - \frac{1}{32} + \frac{1}{243} = 1 - 0.03125 + 0.00412 \approx 0.97287. \]
Boxed answer: $s_3 \approx 0.97287$; error is at most $b_4 = 1/1024 < 0.001$.
applying the Alternating Series Test when terms are not eventually decreasing. The test requires each $b_{n+1} \leq b_n$. Sometimes a sequence decreases only after some index $N$; this is fine -- you need eventual decrease. But for the series $\sum (-1)^n b_n$ where $b_n$ increases, the test does not apply, and checking $b_n \to 0$ alone is not enough to conclude convergence.
Common Errors Summary
| Error | Correction |
|---|---|
| Forgetting to check $b_n \to 0$ | Always verify both conditions; failing $b_n \to 0$ means the series diverges |
| Forgetting to check decreasing | Verify $b_{n+1} \leq b_n$ (or use calculus: if $f(x)$ is decreasing and $b_n = f(n)$) |
| Using the estimation theorem for a non-alternating series | The bound $|R_n| \leq b_{n+1}$ applies only to alternating series satisfying both conditions |
Common Misconceptions
an alternating series converges whenever its terms go to zero.
This is the concept-image-conflicts-definition error about the alternating series test conditions. Both conditions are required: the terms must be decreasing in absolute value AND approach zero. For a sequence like $b_n = 1 + (-1)^n/n$ (which does go to $1$ on average but oscillates), the alternating series test does not apply because $b_n$ is not monotonically decreasing. If $b_n \not\to 0$, the series diverges immediately by the Divergence Test regardless of alternating signs.
the alternating series estimation theorem applies to any series with an error term.
This is the concept-image-conflicts-definition error about the scope of the estimation bound $|R_n| \le b_{n+1}$. This bound holds only for alternating series that satisfy both conditions of the Alternating Series Test. Applying the bound to a non-alternating series or to an alternating series whose terms do not decrease produces an incorrect (too optimistic) error estimate.
Leveled Practice
Level 1
Problem 1. Does $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$ converge?
Show answer
$b_n = 1/\sqrt{n}$: decreasing and $\to 0$. Both conditions hold. The series converges by the Alternating Series Test.
Problem 2. How many terms of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$ are needed to estimate the sum to within $0.01$?
Show answer
Need $b_{n+1} = 1/(n+1)^2 \leq 0.01$, i.e. $(n+1)^2 \geq 100$, so $n+1 \geq 10$, meaning $n \geq 9$. Nine terms suffice.
Mastery Checklist
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