How to Recognize a PSeries
An important type of series is called the pseries. A pseries can be either divergent or convergent, depending on its value. It takes the following form:
Here’s a common example of a pseries, when p = 2:
Here are a few other examples of pseries:
Remember not to confuse pseries with geometric series. Here’s the difference:

A geometric series has the variable n in the exponent — for example,

A pseries has the variable in the base — for example
As with geometric series, a simple rule exists for determining whether a pseries is convergent or divergent.
A pseries converges when p > 1 and diverges when p < 1.
Here are a few important examples of pseries that are either convergent or divergent.
When p = 1: the harmonic series
When p = 1, the pseries takes the following form:
This pseries 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 pseries 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 pseries 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 nth term equals
So this pseries includes every term in the harmonic series plus many more terms. Because the harmonic series is divergent, this series is also divergent.