# Alternating Series Test: Conditions, Error Bound and Examples

> The alternating series test (Leibniz test): if bₙ decreases to 0, then the series of (−1)ⁿbₙ converges. Conditions, the error bound and examples.

Source: https://practicecalculus.com/rules/alternating-series-test
Updated: 2026-11-03

$$ b_n \ge b_{n+1} > 0,\ \lim_{n\to\infty} b_n = 0 \;\Longrightarrow\; \sum (-1)^{n+1} b_n \text{ converges} $$

## At a glance

- **Also called:** The Leibniz test
- **Condition 1:** The terms alternate in sign: $\sum (-1)^{n+1} b_n$ with $b_n > 0$
- **Condition 2:** $b_n$ is decreasing, at least eventually
- **Condition 3:** $\lim_{n\to\infty} b_n = 0$
- **Error bound:** $\lvert S - S_N\rvert \le b_{N+1}$
- **Cannot prove:** Divergence, or absolute convergence

## Key takeaways

- An alternating series converges if the sizes $b_n$ decrease and approach $0$.
- The error after $N$ terms is at most the next term: $\lvert S - S_N\rvert \le b_{N+1}$.
- Passing the test can mean conditional convergence, as with $\sum \frac{(-1)^{n+1}}{n}$.
- If $b_n$ does not go to $0$, the series diverges by the nth-term test.

The **alternating series test** handles series whose terms switch sign, like $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$. If the sizes of the terms shrink steadily to zero, the partial sums zigzag ever more tightly around a limit, and the series converges.

## The test

Suppose $b_n > 0$ for all $n$. The series

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

converges if both:

1. $b_{n+1} \le b_n$ for all $n$ (or for all $n$ past some point), and
2. $\lim_{n\to\infty} b_n = 0$.

The same holds for $\sum (-1)^n b_n$, which only flips every sign.

## The error bound

If the conditions hold and $S$ is the sum, the partial sum $S_N$ is within one term of the truth:

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

Moreover, $S$ lies between any two consecutive partial sums. This is the **alternating series estimation theorem**, and it makes alternating series easy to approximate.

```graph
title: 'Partial sums of the alternating harmonic series zigzag toward ln 2'
caption: 'Odd partial sums land above $\ln 2 \approx 0.693$ and even ones below, with the gap shrinking each step.'
x: [0, 9]
y: [0.4, 1.1]
functions:
  - expr: ln(2)
    label: y = ln 2
    dashed: true
points:
  - [1, 1, 'S1 = 1']
  - [2, 0.5, 'S2 = 0.5']
  - [3, 0.8333]
  - [4, 0.5833]
  - [5, 0.7833]
  - [6, 0.6167]
  - [7, 0.7595]
  - [8, 0.6345]
```

## Why it works

Because the $b_n$ decrease, each step overshoots less than the previous one. The even partial sums $S_2, S_4, S_6, \ldots$ increase, the odd ones $S_1, S_3, S_5, \ldots$ decrease, and every even sum stays below every odd sum. Both sequences are bounded and monotone, so they converge, and since $S_{2k+1} - S_{2k} = b_{2k+1} \to 0$, they converge to the same limit.

## Worked examples

**Example 1: the alternating harmonic series.** For $\sum \frac{(-1)^{n+1}}{n}$, $b_n = \frac{1}{n}$ is positive, decreasing and tends to $0$. The series converges (to $\ln 2$). Since $\sum \frac{1}{n}$ diverges, the convergence is **conditional**.

**Example 2: failing the limit condition.** For $\sum (-1)^n \frac{n}{n+1}$, $b_n = \frac{n}{n+1} \to 1 \ne 0$. The alternating series test does not apply, and the [nth-term test](/rules/nth-term-test) shows the series diverges.

**Example 3: checking "decreasing" with a derivative.** For $\sum (-1)^{n+1}\frac{\ln n}{n}$ (starting at $n = 2$), let $f(x) = \frac{\ln x}{x}$. Then $f'(x) = \frac{1 - \ln x}{x^2} < 0$ for $x > e$, so $b_n$ decreases for $n \ge 3$, and $\frac{\ln n}{n} \to 0$. The series converges.

**Example 4: estimating a sum.** The series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n!} = 1 - \frac{1}{2} + \frac{1}{6} - \frac{1}{24} + \cdots$ satisfies the test. Four terms give

$$S_4 = 1 - \frac{1}{2} + \frac{1}{6} - \frac{1}{24} = \frac{5}{8} = 0.625$$

with error at most $b_5 = \frac{1}{120} \approx 0.0083$. The true sum is $1 - \frac{1}{e} \approx 0.6321$, and the actual error is about $0.0071$.

## Absolute vs conditional convergence

The test proves convergence but says nothing about absolute convergence. Check $\sum b_n$ separately:

| Series | $\sum b_n$ | Verdict |
| --- | --- | --- |
| $\sum \frac{(-1)^{n+1}}{n^2}$ | converges | absolutely convergent |
| $\sum \frac{(-1)^{n+1}}{n}$ | diverges | conditionally convergent |
| $\sum \frac{(-1)^{n+1}}{\sqrt{n}}$ | diverges | conditionally convergent |

The [ratio test](/rules/ratio-test) gives $L = 1$ for all three, so it cannot settle any of them.

## Common mistakes

1. **Using the test to prove divergence.** Failing the decreasing condition does not mean divergence. Only $b_n \not\to 0$ does, and that is the nth-term test.
2. **Including the sign in $b_n$.** $b_n$ is the positive size of the term, with $(-1)^n$ removed.
3. **Skipping the decreasing check.** It is usually true, but must be verified, sometimes with a derivative.
4. **Calling a series absolutely convergent** because it passed this test.

For a comparison of every convergence test and how to pick one, read the full guide to [convergence tests](/blog/convergence-tests).

## Frequently asked questions

### What are the conditions of the alternating series test?

The series must have the form $\sum (-1)^{n+1} b_n$ (or $\sum (-1)^n b_n$) with $b_n > 0$, the $b_n$ must be decreasing (at least eventually), and $\lim b_n = 0$. If all three hold, the series converges.

### Does the alternating harmonic series converge?

Yes. $1 - \frac{1}{2} + \frac{1}{3} - \cdots$ passes the test and converges to $\ln 2$. It converges only conditionally, since $\sum \frac{1}{n}$ diverges.

### What is the alternating series error bound?

If a series satisfies the test, stopping after $N$ terms gives an error no larger than the first omitted term: $\lvert S - S_N\rvert \le b_{N+1}$.

### What if an alternating series fails the test?

If $b_n \not\to 0$ the series diverges by the nth-term test. If only the decreasing condition fails, the test is inconclusive and you need another method.