Topic 10.9 Notes – Determining Absolute or Conditional Convergence

Absolute, Conditional, or Divergent

For a series $\sum a_n$, there are three possible outcomes:

1. Absolutely convergent
2. Conditionally convergent
3. Divergent

Here’s what each one actually means.

Absolutely Convergent

A series is absolutely convergent if

$$\sum |a_n| \text{ converges.}$$

That means even after removing all signs, the series still converges.

Two facts you need locked in:

• If a series is absolutely convergent, then it converges.
• Rearranging or regrouping its terms does not change the sum.

Absolute convergence is the “strongest” kind. It’s stable and predictable.

Conditionally Convergent

A series is conditionally convergent if:

• $\sum a_n$ converges,
• but $\sum |a_n|$ diverges.

This almost always shows up with alternating series.

Here’s the key idea: the series converges only because the positive and negative terms cancel in a delicate way.

Important consequence:

• Rearranging the terms can actually change the sum.

Also, wording matters on tests. If you just say “convergent,” that implies absolute convergence. If it’s conditional, say so clearly.

Divergent

A series is divergent if $\sum a_n$ does not converge.

If the original series diverges, you’re done. No classification beyond that.

The Decision Process You Should Use

When you’re asked to classify a series, the order matters.

Step 1: Check Absolute Convergence

Look at

$$\sum |a_n|$$

Then apply any test you know:

• Comparison or Limit Comparison
• p-series recognition
• Ratio Test
• Root Test
• Integral Test

If $\sum |a_n|$ converges →
✔️ Absolutely convergent (and therefore convergent). Stop.

Step 2: If Absolute Convergence Fails

If $\sum |a_n|$ diverges:

Now test the original series $\sum a_n$.

Often this means using the Alternating Series Test (AST).

• If the original converges → ✔️ Conditionally convergent
• If the original diverges → ✔️ Divergent

This two-step structure is exactly how FRQs expect your reasoning to flow.

The Alternating Series Situation

Most conditional convergence problems look like:

$$\sum (-1)^n b_n \quad \text{or} \quad \sum (-1)^{n+1} b_n$$

The Alternating Series Test requires:

1. $b_n > 0$
2. $b_n$ is decreasing
3. $\lim_{n\to\infty} b_n = 0$

If all three are true, the series converges.

Now here’s the crucial move:

• Take the absolute value.
• The alternating sign disappears.
• You’re left testing $\sum b_n$.

If that new series diverges (for example, it behaves like $1/n$), then the original is conditionally convergent.

If it converges (like a $p$-series with $p>1$), then the original was actually absolutely convergent the whole time.

The Ratio or Root Test often confirms absolute convergence quickly, especially when factorials or exponentials are involved.

When $L < 1$, you immediately know the series is absolutely convergent. When $L > 1$, it diverges. If $L = 1$, the test does not decide, so you try something else.

Example Patterns You Should Recognize

Example 1

$$\sum \frac{(-1)^n}{n^{1/2}}$$

• Absolute value gives $\sum \frac{1}{n^{1/2}}$, a $p$-series with $p=\frac12$, which diverges.
• Alternating Series Test works.

✔️ Conditionally convergent

Example 2

$$\sum \frac{\cos n}{n^4}$$

Use the inequality $|\cos n| \le 1$.

$$\left|\frac{\cos n}{n^4}\right| \le \frac{1}{n^4}$$

Since $\sum 1/n^4$ converges, the original is:

✔️ Absolutely convergent

Notice this one isn’t alternating in a clean $(-1)^n$ way. Absolute convergence doesn’t require neat sign patterns.

Why Absolute Convergence Is Stronger

Here’s the big structural idea.

If a series is absolutely convergent:

• It converges.
• Any rearrangement has the same sum.

If a series is only conditionally convergent:

• Rearranging can change the value.
• The convergence depends on cancellation.

That rearrangement fact is explicitly testable. It’s not trivia.