# 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.

**Table of Contents**

## 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.

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