P-Series: Definition & Examples

Instructor: Cameron Smith

Cameron has a Master's Degree in education and has taught HS Math for over 25 years.

This lesson is designed to help you understand a specific type of series called a p-series. You will determine if a series is a p-series, and you will learn to decide if a p-series converges or diverges.

Definition of a p-Series

A p-series is a specific type of infinite series. It is a series of the form

definition

where p can be any real number greater than zero.

Notice that in this definition n will always take on positive integer values, and the series is an infinite series because it is a sum containing infinite terms.

There are infinitely many p-series because you have infinite choices for p. Each time you choose a different value for p you create another p-series.

When working with infinite series, you will want to know if they converge or diverge. With p-series, if p > 1, the series will converge, or in other words the series will add up to a specific numerical value. If 0 < p ≤ 1, the series will diverge, which means that the series will not add up to a specific numerical value.

Examples of Different p-Series

One of the basic p-series is called the harmonic series. This is the series that is formed when p = 1. It looks like this:

harmonic series

The harmonic series is used a lot in Calculus in comparisons with other series; it is also used in tests for convergence and divergence.

Let's take a look at some more p-series. Here is what the p-series looks like when p = 2:

Example 2

As you can see, there are many p-series, and p doesn't always have to be a whole number. For example, here is the p-series when p = 1/2:

example 4

Look closely at the preceding example. When 0 < p < 1, series such as this include all the terms of the harmonic series plus a lot of terms in between. Since the harmonic series diverges, these series that include all the terms of the harmonic series, plus other terms, will also diverge.

Why Can't p Be Negative?

If you make p a negative number, the same outcome will happen every time. For example if we let p = -1, the series would look like this:

Example 5

You can see in this example, when p = -1 the value of each term in the sequence is increasing. Therefore the series is obviously diverging, since you are adding larger and larger values to the sum.

Let's look at what would happen if we let p be another negative number, p = -3/2.

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