High School Math Solutions - Series Convergence Calculator, p-Series Test

Last blog post, we discussed what an infinite series is and how to determine if an infinite series converges using the geometric series test. In this blog post, we will discuss how to determine if an infinite series converges using the p-series test.

A p-series is a series of the form∑_{n=1}^∞\frac{1}{n^p} , where p is a constant power.

Here is an example of a p-series:

1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+ ...=\frac{1}{1^2} +\frac{1}{2^2} +\frac{1}{3^2} +\frac{1}{4^2} + ...=∑_{n=1}^∞\frac{1}{n^2}

So, how do we determine if the sum of a p-series converges to a finite number or diverges to an infinite number? We use the p-series test!

The following is the p-series test:

If the series is of the form ∑_{n=1}^∞\frac{1}{n^p}   , where p>0, then
If p>1, then the series converges.
If 0≤p<1, then the series diverges.

Unlike the geometric test, we are only able to determine whether the series diverges or converges and not what the series converges to, if it converges.

The p-series test is fairly simple, useful, and easy to remember.

Let’s see some examples of how to use it.

First example (click here):

                                                          ∑_{n=1}^∞\frac{1}{\sqrt{n}}
1. Determine the value of p

                                                ∑_{n=1}^∞\frac{1}{\sqrt{n}}= ∑_{n=1}^∞\frac{1}{n^{\frac{1}{2}}}

                                                                 p=  \frac{1}{2}

2. Determine whether the series converges or diverges

                              Since p=  \frac{1}{2} and therefore 0≤p<1, the series diverges.

Next example (click here):

                                                              ∑_{n=1}^∞\frac{n^2}{n^6}

1. Determine the value of p

                                     ∑_{n=1}^∞\frac{n^2}{n^6}   = ∑_{n=1}^∞\frac{1}{n^{6-2}} = ∑_{n=1}^∞\frac{1}{n^4}

                                                                    p=4

       In this step, I used the following exponent rule: \frac{x^a}{x^b} =\frac{1}{x^{b-a}}

2. Determine whether the series converges or diverges

                                      Since p=4 and therefore p>1, the series converges.

Last example (click here):

                                              ∑_{n=1}^∞\frac{cos^2(n)+sin^2(n)}{n^2}

1. Determine the value of p

                                    ∑_{n=1}^∞\frac{cos^2(n)+sin^2(n)}{n^2} = ∑_{n=1}^∞\frac{1}{n^2}

                                                                   p=2

In this step, I used the following trigonometric identity: sin^2(x)+cos^2(x)=1

2. Determine whether the series converges or diverges

                                  Since p=4 and therefore p>1, the series converges.

The p-series test is pretty straightforward, helpful, and not too difficult. For more help or practice on the p-series test, check out Symbolab’s Practice. Next blog post, I’ll go over the convergence test for alternating series.

Until next time,

Leah