Alternating Series Test for MATH 141

Understanding the Alternating Series Test

So far, all our convergence tests have required positive terms. But what about series that alternate between positive and negative? The Alternating Series Test gives us a surprisingly simple way to determine convergence for these series — and often, series that would diverge with all positive terms will converge once you add alternating signs.

What is an Alternating Series?

An alternating series has terms that flip between positive and negative:

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

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

where $b_n > 0$ for all $n$.

Classic examples:

• $\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ (alternating harmonic)
• $\displaystyle -1 + \frac{1}{4} - \frac{1}{9} + \frac{1}{16} - \cdots$

The Test (Two Conditions)

The alternating series $\displaystyle\sum_{n=1}^{\infty} (-1)^{n+1} b_n$ converges if:

Condition 1: $b_{n+1} \leq b_n$ for all $n$ (decreasing)

Condition 2: $\displaystyle\lim_{n \to \infty} b_n = 0$

Both conditions must be satisfied!

Why Does This Work?

The partial sums $S_n$ "bounce" back and forth on the number line, getting closer to the limit $S$ with each term.

Starting from 0, add $b_1$ to get $S_1$. Then subtract $b_2$ to get $S_2$. Then add $b_3$ to get $S_3$, and so on. Each step moves you back and forth: $S_1$, $S_3$, $S_5$, ... approach $S$ from the right while $S_2$, $S_4$, $S_6$, ... approach from the left. Because the $b_n$ are decreasing toward zero, the bounces get smaller, trapping $S$ in an ever-shrinking interval.

Visualizing Terms and Partial Sums

This shows both the terms $a_n = (-1)^{n+1}b_n$ and partial sums $S_n$ plotted against $n$:

The blue dots show the terms $a_n$ alternating above and below zero, with magnitudes shrinking toward zero. The red dots show the partial sums $S_n$ oscillating but converging toward the series sum. Odd partial sums approach from above; even partial sums approach from below.

The Alternating Series Estimation Theorem

If an alternating series satisfies both conditions of the test, then:

$$|S - S_n| \leq b_{n+1}$$

The error in approximating $S$ with $S_n$ is at most the first omitted term.

Absolute vs Conditional Convergence

The alternating harmonic series is conditionally convergent: it converges, but its absolute value (the harmonic series) diverges.

Common Mistakes and Misunderstandings

❌ Mistake: Forgetting to check both conditions

Wrong: "The terms go to zero, so it converges."

Why it's wrong: Both conditions are required. Terms must decrease AND approach zero.

Correct: Always verify $b_{n+1} \leq b_n$ AND $\lim b_n = 0$.

❌ Mistake: Confusing $a_n$ and $b_n$

Wrong: Working with $a_n = \frac{(-1)^{n+1}}{n}$ directly when checking conditions.

Why it's wrong: The conditions apply to $b_n = |a_n|$, the positive part.

Correct: Extract $b_n$ first: if $a_n = \frac{(-1)^{n+1}}{n}$, then $b_n = \frac{1}{n}$.

❌ Mistake: Assuming alternating = converges

Wrong: "It alternates, so it must converge."

Why it's wrong: Not all alternating series converge! See Example 3.

Correct: Apply the test — both conditions must hold.