# How to Recognize a P-Series

An important type of series is called the *p*-series. A *p-*series can be either divergent or convergent, depending on its value. It takes the following form:

Here’s a common example of a *p-*series, when *p* = 2:

Here are a few other examples of *p*-series:

Remember not to confuse *p*-series with geometric series. Here’s the difference:

A geometric series has the variable

*n*in the exponent — for example,A

*p*-series has the variable in the base — for example

As with geometric series, a simple rule exists for determining whether a *p*-series is convergent or divergent.

A *p*-series converges when *p* > 1 and diverges when *p* < 1.

Here are a few important examples of *p*-series that are either convergent or divergent.

## When *p* = 1: the harmonic series

When *p *= 1, the *p*-series takes the following form:

This *p*-series is important enough to have its own name: the *harmonic series.* The harmonic series is *divergent.*

## When *p* = 2, *p* = 3, and *p* = 4

Here are the *p*-series when *p *equals the first few counting numbers greater than 1:

Because *p *> 1, these series are all *convergent**.*

## When *p* = 1/2

When *p* = 1/2 the *p*-series looks like this:

Because *p **≤* 1, this series *diverges.* To see why it diverges, notice that when *n* is a square number, say *n* = *k*^{2}, the *n*th term equals

So this *p*-series includes every term in the harmonic series plus many more terms. Because the harmonic series is divergent, this series is also divergent.