P-Series: Definition and P-Series Test Calculator

The p-series is useful in calculus because it can be used to test for convergence and divergence of other series.

p-Series Definition

In calculus and real analysis, a p-series is a series of the form:

$$ \displaystyle{\sum_{n=1}^\infty} \dfrac{1}{n^p} = 1+\dfrac{1}{2^p}+ \dfrac{1}{3^p} + \dfrac{1}{4^p} + \ldots + \dfrac{1}{n^p}+\ldots$$

or, $$\displaystyle{\sum_{n=1}^\infty} n^{-p}= 1+2^{-p}+ 3^{-p} + 4^{-p} + \ldots + n^{-p}+\ldots$$

here, $n$ is a positive integer and $p$ is a positive real number $\iff n \in \mathbf{N^+}$ and $p \in \mathbb{R}$

When p > 1, the terms of the series get smaller and smaller as n gets larger, and the series converges.

This convergence is known as convergence in the "ordinary sense" because the series approaches a finite limit as the number of terms increases.

When p ≤ 1, the terms of the series do not get smaller as n gets larger, and the series diverges. This divergence means that the series does not approach a finite limit as the number of terms increases.

p-Series Test of Convergence

The p-series is useful in calculus because it can be used to test for convergence and divergence of other series. Specifically, if a series can be compared and shown as equivalent to a p-series with p > 1, then the series converges. Conversely, if a series can be shown to be equivalent to a p-series with p ≤ 1, then the series diverges.

p-Series Test Calculator and Grapher

While there are mathematical ways to use the p-series test to check if a series is convergent or not, I have created a p-series test calculator and grapher that can help you do this online.

Please note that I have taken the $\displaystyle{\sum_{n=1}^\infty} n^{-p}= 1+2^{-p}+ 3^{-p} + 4^{-p} + \ldots + n^{-p}+\ldots$ form, which is same as the $\dfrac{1}{n^p}$ form you may have been seeing in your books.

Click on the button above to launch the p-Series Test Calculator and Grapher. If the button doesn't appear then you are using an incompatible browser or your browser doesn't have proper Javascript support. Try opening this page in Google Chrome or any other modern browser. This works on mobile devices too.