Topic 10.10 Notes – Alternating Series Error Bound

1. The Alternating Series Error Bound

We’re working with alternating series of the form

$$\sum_{n=1}^\infty (-1)^n a_n \quad \text{or} \quad \sum_{n=1}^\infty (-1)^{n-1} a_n$$

where the signs switch back and forth.

Before you can use the error bound, the series must converge by the Alternating Series Test (AST). That means:

• $a_n > 0$
• $a_n$ is decreasing
• $\lim_{n\to\infty} a_n = 0$

If those are true, then not only does the series converge, but we also get this:

$$|S - S_N| \le a_{N+1}$$

Where:

• $S$ = the true infinite sum
• $S_N$ = the sum of the first $N$ terms
• $a_{N+1}$ = the first omitted term (without the alternating sign)

That’s the whole theorem. The error is at most the size of the next term.

And this only works for alternating series that pass AST.

2. How to Use the Error Bound

There are three ways this shows up on quizzes and FRQs.

A. “Find the error bound after N terms”

Suppose

$$\sum_{n=1}^\infty (-1)^{n-1} \frac{1}{n^2}$$

Using 4 terms means the first omitted term is:

$$a_5 = \frac{1}{5^2} = \frac{1}{25}$$

So:

$$|S - S_4| \le \frac{1}{25}$$

That fraction is your error bound. You don’t even need the partial sum unless they ask for it.

B. “Approximate the sum and give an interval”

Say you compute $S_3 = 1 - \frac14 + \frac19$.

The next term is $a_4 = \frac{1}{16}$.

So:

$$|S - S_3| \le \frac{1}{16}$$

Turn that into an interval:

$$S_3 - \frac{1}{16} \le S \le S_3 + \frac{1}{16}$$

That interval guarantees the true sum is inside.

On FRQs, writing the inequality clearly usually earns the point.

C. “How many terms for error < 0.001?”

This is the one that trips people up.

You use:

$$a_{N+1} < 0.001$$

If $a_n = \frac{1}{n^2}$, then:

$$\frac{1}{(N+1)^2} < 0.001$$

Solve it:

$$(N+1)^2 > 1000$$

$$N+1 > \sqrt{1000} \approx 31.6$$

$$N+1 = 32 \Rightarrow N = 31$$

Always round up. You’re guaranteeing the error is small enough.

Notice you didn’t compute any partial sums. You only needed the next term.

3. What’s Actually Happening

Alternating partial sums zig-zag toward the true value.

Here’s the picture to have in your head as you think about the alternating harmonic series:

Partial sums of the alternating harmonic series approaching ln 2

Each partial sum lands on opposite sides of the true sum, shown by the horizontal line.

Because:

• Signs alternate
• Terms get smaller

The next term represents the biggest possible overshoot. Once terms shrink, the error automatically shrinks.

That’s why the first omitted term controls everything.

Also, the true sum always lies between two consecutive partial sums. That’s a common AP multiple choice idea.

4. Common Exam Traps

• Forgetting to verify AST first. If it’s not decreasing or the limit isn’t zero, the theorem doesn’t apply.
• Using $a_N$ instead of $a_{N+1}$. If you use $S_6$, the bound uses the 7th term.
• Keeping the sign. The error bound is positive. Use the absolute value of the next term.
• Rounding down when solving for N. That ruins the guarantee.
• Mixing this up with Taylor polynomial error bounds. Totally different theorem.

On the AP exam, they often combine this with a “justify convergence” step first. If you don’t state decreasing and limit zero, you can lose points even if your bound is correct.