Alternating Series
students, imagine watching a swinging sign outside a store 🪧. It moves one way, then the other way, and each swing gets smaller. Many alternating series behave in a similar pattern: the terms switch signs, and the sizes of the terms shrink over time. In Calculus 2, this idea matters because some series that do not have all positive terms can still converge.
In this lesson, you will learn how to recognize alternating series, test whether they converge, and understand why the terms must shrink toward zero. By the end, you should be able to explain the main ideas, use the Alternating Series Test, and connect this topic to the larger set of convergence tests in Calculus 2.
What Is an Alternating Series?
An alternating series is an infinite series whose terms switch signs back and forth. A common form is
$$\sum_{n=1}^{\infty} (-1)^{n-1} a_n$$
or
$$\sum_{n=1}^{\infty} (-1)^n a_n$$
where $a_n \ge 0$ for every $n$.
The key idea is that the sign alternates, but the sequence $a_n$ gives the size of each term. For example,
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$$
is alternating because the signs go positive, negative, positive, negative, and so on. Here the sizes are $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and so on.
A lot of real-world situations involve changes in direction or corrections. For example, a balance scale might overshoot one way, then correct back the other way, with each correction getting smaller. Alternating series model this back-and-forth behavior well 📉📈.
The Alternating Series Test
The main tool for this topic is the Alternating Series Test, often called the AST. It says that the series
$$\sum_{n=1}^{\infty} (-1)^{n-1} a_n$$
converges if both of these conditions are true:
- $a_n$ is decreasing eventually, meaning $a_{n+1} \le a_n$ for all large enough $n$.
- $\lim_{n\to\infty} a_n = 0$.
The word “eventually” matters because the terms do not need to decrease from the very first term. They just need to decrease after some point.
This test is powerful because it gives convergence even when other tests for positive-term series do not apply directly. The alternating signs can create cancellation, which helps the partial sums settle down.
Let’s look at an example:
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$
Here $a_n = \frac{1}{n}$. The sequence $\frac{1}{n}$ decreases, and
$$\lim_{n\to\infty} \frac{1}{n} = 0$$
so the series converges by the Alternating Series Test. This famous series is called the alternating harmonic series.
Why the Terms Must Go to Zero
The condition
$$\lim_{n\to\infty} a_n = 0$$
is essential. If the terms do not go to zero, the series cannot converge. This is true for any infinite series, not just alternating ones.
For example,
$$\sum_{n=1}^{\infty} (-1)^{n-1}$$
has terms $1, -1, 1, -1, \dots$ . The terms do not approach $0$, so the series diverges. In fact, the partial sums keep bouncing between $1$ and $0$:
$$1,\ 0,\ 1,\ 0,\dots$$
Because the partial sums do not approach a single number, the series diverges.
This shows an important rule: alternating signs alone do not guarantee convergence. The sizes of the terms must shrink to zero. Think of it like a basketball bouncing. If each bounce stays the same height forever, it never settles. But if each bounce gets smaller and smaller, the ball eventually comes to rest 🏀.
How to Use the Alternating Series Test
To apply the AST, students, follow these steps:
Step 1: Identify $a_n$
Rewrite the series so the sign pattern is separated from the positive part. For example,
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2+1}$$
has $a_n = \frac{1}{n^2+1}$.
Step 2: Check that $a_n$ decreases eventually
You want to know whether the positive terms get smaller as $n$ grows. For many common formulas, this can be done by comparing values or using calculus reasoning.
For the example above, $\frac{1}{n^2+1}$ gets smaller as $n$ increases because the denominator gets larger.
Step 3: Check the limit
Compute
$$\lim_{n\to\infty} a_n$$
If the limit is $0$, and the terms decrease eventually, then the alternating series converges.
For the same example,
$$\lim_{n\to\infty} \frac{1}{n^2+1} = 0$$
so the series converges by the AST.
Example: Convergent Alternating Series
Consider
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$$
Here $a_n = \frac{1}{\sqrt{n}}$.
- The sequence $a_n$ decreases as $n$ increases.
- The limit is
$$\lim_{n\to\infty} \frac{1}{\sqrt{n}} = 0$$
So the series converges by the Alternating Series Test.
But here is an important detail: this series does not converge absolutely, because
$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$$
is a $p$-series with $p = \frac{1}{2}$, and since $p \le 1$, it diverges. That means the original alternating series converges only conditionally.
This distinction matters in Convergence Tests II because a series can converge in more than one way:
- Absolute convergence means the series of absolute values converges.
- Conditional convergence means the alternating series converges, but the absolute-value series does not.
Example: Divergent Alternating Series
Now consider
$$\sum_{n=1}^{\infty} (-1)^{n-1} \frac{n}{n+1}$$
Here
$$a_n = \frac{n}{n+1}$$
and
$$\lim_{n\to\infty} \frac{n}{n+1} = 1$$
Since the limit is not $0$, the series diverges. Even though the signs alternate, the terms do not shrink to zero. The AST cannot be used to prove convergence here, and in fact the series must diverge.
This example shows why checking the limit is not optional. If the limit test fails, the alternating series test fails too.
Remainder Estimate and Accuracy
One useful feature of alternating series is that they let us estimate the error when we stop after a certain number of terms. If a series satisfies the Alternating Series Test, then the error after $N$ terms is at most the size of the next term.
If $S$ is the full sum and $S_N$ is the $N$th partial sum, then
$$|S-S_N| \le a_{N+1}$$
This is called the alternating series remainder estimate.
For example, if you approximate
$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$
using the first $5$ terms, the error is at most
$$\frac{1}{6}$$
That gives a practical way to control accuracy. In real life, this is useful when a computer or calculator uses only a few terms to estimate a value. The next term tells you how far off you might be.
How Alternating Series Fit Into Convergence Tests II
Alternating series are part of the bigger family of convergence tests in Calculus 2. In this unit, you often decide which test fits the structure of the series:
- The Ratio Test works well for factorials and powers.
- The Root Test is often useful when terms look like something raised to the $n$th power.
- The Alternating Series Test is best when the signs switch back and forth.
Sometimes one series can be examined by more than one test. For instance, if you have an alternating series, you may first use the AST to check convergence, and then separately check whether it converges absolutely using another test.
This is why alternating series are important in the larger course: they show that not all convergence comes from positive terms shrinking smoothly. Cancellation from alternating signs can also produce convergence.
Conclusion
students, alternating series are infinite series whose signs switch back and forth. The Alternating Series Test says that such a series converges if the positive part $a_n$ eventually decreases and
$$\lim_{n\to\infty} a_n = 0$$
This lesson showed that alternating signs alone are not enough; the term sizes must shrink to zero. You also learned that alternating series can converge conditionally, and that the remainder estimate gives a useful bound on error.
In the bigger picture of Convergence Tests II, alternating series give you a powerful way to analyze series that are not handled well by tests for only positive terms. When you see signs switching back and forth, think about the AST, the limit, and whether the terms are getting smaller and smaller.
Study Notes
- An alternating series has terms whose signs switch back and forth.
- A common form is $\sum_{n=1}^{\infty} (-1)^{n-1} a_n$ with $a_n \ge 0$.
- The Alternating Series Test says the series converges if:
- $a_n$ decreases eventually, and
- $$\lim_{n\to\infty} a_n = 0$$
- If the terms do not go to zero, the series diverges.
- Alternating series may converge conditionally, meaning the series converges but the absolute-value series does not.
- For a convergent alternating series, the error after $N$ terms satisfies $|S-S_N| \le a_{N+1}$.
- Alternating series are one part of Convergence Tests II, alongside the Ratio Test and Root Test.