AP Calculus BC

Geometric Series vs p-Series

Q: Is the harmonic series geometric or a p-series?

It is the p-series with p = 1, sitting exactly on the boundary, and it diverges.

A geometric series has a constant ratio between consecutive terms, with the index in the exponent, and converges when the absolute ratio is below one. A p-series has the index in the base raised to a fixed power and converges when that power exceeds one.

Geometric series

Use when: Each term is a fixed multiple of the previous one, so the index sits in the exponent.

p-series

Use when: The terms are reciprocals of a power of the index, so the index sits in the base.

Side by side

	Geometric	p-series
Form	$\sum ar^n$	$\sum \frac{1}{n^p}$
Index appears in	The exponent	The base
Converges when	$\lvert r \rvert < 1$	$p > 1$
Sum is known	Yes, $\frac{a}{1-r}$	No

Both are classified at sight, which is why they are the comparison targets for every other test. Recognising which one you are looking at is a matter of asking whether $n$ is upstairs or downstairs.

The geometric series is one of very few whose exact sum you can write down. A p-series tells you only whether it converges, not to what, with the single exception of specific known results outside the AP scope.

The geometric trap

In $\frac{a}{1-r}$ the value $a$ is the first term actually present. A sum starting at $n = 3$ has a different first term from one starting at $n = 0$ .

Frequently asked questions

Is the harmonic series geometric or a p-series?

It is the p-series with $p = 1$ , sitting exactly on the boundary, and it diverges.

In the CED: Unit 10: Infinite Sequences and Series (BC)