Topic 10.5 Notes – Harmonic Series and p-Series

Harmonic Series and p-Series

A p-series has the form

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

where $p$ is a constant.

All terms are positive, so this is a positive-term series. The only question is whether the infinite sum converges or diverges.

The p-Series Rule

$$\text{If } p > 1, \text{ the series converges.}$$

$$\text{If } p \le 1, \text{ the series diverges.}$$

That’s the entire test. No partial sums. No derivatives. Just identify $p$.

The Harmonic Series

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

This is the p-series with $p = 1$.

Since $p = 1 \le 1$, it diverges.

This one matters a lot. It’s the classic example that shows:

• Even if $a_n \to 0$,
• The series can still diverge.

On quizzes and the AP exam, you’re expected to recognize this instantly.

Why the Value $p = 1$ Is the Boundary

Here’s the key behavior. Compare the graphs of $y = \frac{1}{x}$ and $y = \frac{1}{x^2}$:

Graphs of $y = 1/x$ and $y = 1/x^2$

Both functions go to 0 as $x \to \infty$.

• $\frac{1}{x}$ decreases slowly → harmonic series diverges
• $\frac{1}{x^2}$ decreases faster → converges

The exponent controls how fast the terms shrink. If they don’t shrink fast enough, the total sum keeps growing without bound.

That’s the intuition behind the rule.

How to Recognize a p-Series

Most mistakes happen because students don’t simplify first.

You must rewrite the general term so it looks exactly like:

$$\frac{1}{n^p}$$

Simplifying Correctly

Common rewrites:

• $n^{-4} = \frac{1}{n^4}$
• $\frac{n^3}{n^5} = \frac{1}{n^2}$
• $\frac{1}{\sqrt{n}} = \frac{1}{n^{1/2}}$
• $\frac{1}{n^{0.6}\sqrt{n}} = \frac{1}{n^{1.1}}$

Example:

$$\sum n^{-2.5}$$

Rewrite as:

$$\sum \frac{1}{n^{2.5}}$$

Here $p = 2.5 > 1$, so it converges.

Another example:

$$\sum \frac{n^4}{n^5}$$

Simplify:

$$\frac{n^4}{n^5} = \frac{1}{n}$$

That’s the harmonic series → diverges.

Always simplify completely before deciding.

The Cases You Should Instantly Recognize

✔️ $p > 1$ → Converges

• $\sum \frac{1}{n^3}$
• $\sum \frac{1}{n^{1.2}}$
• $\sum \frac{1}{n^{9}}$

Even $p = 1.001$ works.

❌ $p = 1$ → Harmonic → Diverges

• $\sum \frac{1}{n}$

This shows up constantly in comparison tests later.

❌ $0 < p < 1$ → Diverges

• $\sum \frac{1}{n^{0.8}}$
• $\sum \frac{1}{n^{1/3}}$

The terms go to zero, just not fast enough.

❌ $p \le 0$

Examples:

• $\sum 1$
• $\sum n$
• $\sum n^2$

These terms don’t even approach 0. By the nth-term test, they diverge immediately.

How This Appears on Tests

You’ll usually see one of three things:

1. Straight recognition

   Determine convergence or divergence.

   Simplify → identify $p$ → apply rule → state conclusion.

2. Disguised p-series

   Expressions with radicals, negative exponents, or rational powers. The work is in the algebra.

3. Used as a benchmark

   Later in comparison tests, you’ll compare to:

   • $\sum 1/n$ (diverges)
   • $\sum 1/n^2$ (converges)

   If you don’t instantly know those two, comparison problems get harder.

Common Errors

• Seeing $n^{-3}$ and thinking $p = -3$. It becomes $1/n^3$, so $p = 3$.
• Ignoring the numerator in rational expressions.
• Believing “terms go to 0” guarantees convergence.
• Forgetting that $p = 1$ diverges.