Topic 10.7 Notes – Alternating Series Test for Convergence

What an Alternating Series Is

An alternating series has terms that change sign: positive, negative, positive, negative, and so on.

Common ways this shows up:

$$\sum (-1)^n a_n, \quad \sum (-1)^{n+1} a_n, \quad \sum \cos(n\pi)\, a_n$$

Since $\cos(n\pi) = (-1)^n$, trig expressions can hide alternating behavior.

When you apply the Alternating Series Test, you rewrite the series in the form

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

where:

• $a_n > 0$
• $a_n$ represents the magnitude only (ignore the sign)

You always analyze the behavior of $a_n$, not the signed term.

The Alternating Series Test

The Alternating Series Test (AST) says:

An alternating series $\sum (-1)^n a_n$ or $\sum (-1)^{n+1} a_n$ converges if:

$$\lim_{n\to\infty} a_n = 0$$

and
$a_n$ is decreasing (eventually decreasing is enough).

That’s it. Two conditions.

What each condition means

• Limit equals 0
  If $\lim a_n \neq 0$, the series diverges immediately by the nth-term test.
  This overrides everything else.
• Decreasing
  You must show $a_{n+1} < a_n$ for large $n$.
  You can:
  • Compare $a_{n+1}$ and $a_n$, or
  • Treat $a_n$ as $f(x)$ and show $f'(x) < 0$

If both conditions hold, you conclude:
“Therefore, the series converges by the Alternating Series Test.”

Important: AST proves convergence only. It does not prove divergence unless the limit condition fails.

Applying the Alternating Series Test

Here’s the clean process you’ll use on quizzes and FRQs:

1. Confirm it alternates
   • Look for $(-1)^n$, $(-1)^{n+1}$, or $\cos(n\pi)$.
2. Identify $a_n$
   Remove the alternating factor and make sure what remains is positive.
3. Check the limit
   • Compute $\lim a_n$.
   • If not 0 → stop. Diverges.
4. Check decreasing behavior
   • Compare successive terms, or
   • Take a derivative.
5. State the test clearly in your conclusion

On free response, skipping the “decreasing” justification costs points. Writing only “limit = 0” is not enough.

Typical Patterns You’ll See

Rational expressions

Example structure:

$$a_n = \frac{n^3}{n^3 + 5}$$

If numerator and denominator have the same degree, the limit is a nonzero constant → diverges immediately.

If the denominator grows faster, the limit is 0 → continue checking decreasing behavior.

Harmonic-type terms

$$a_n = \frac{1}{n}, \quad \frac{1}{n^p}$$

• Limit = 0
• Decreasing

So the alternating version converges, even though the regular harmonic series diverges. That contrast shows up a lot on multiple choice.

Logarithmic terms

$$a_n = \ln(n)$$

Limit goes to infinity → diverges immediately.

$$a_n = \frac{\ln(n)}{n}$$

Limit = 0. Usually decreasing for large $n$, so AST often works here.

Trig disguises

Always simplify:

$$\cos(n\pi) = (-1)^n$$

Students sometimes miss this and use the wrong test. The AP loves hiding alternating behavior inside trig.

Conditional vs Absolute Convergence

AST only tells you the series converges. It does not tell you whether it converges absolutely.

If
$$\sum |a_n|$$
also converges, that’s absolute convergence.

If the alternating series converges but the non-alternating version diverges, that’s conditional convergence.

AST alone does not test absolute convergence.