How To Recognize A P-Series

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HomeAcademics & The Arts ArticlesMath ArticlesCalculus ArticlesHow to Recognize a P-SeriesByMark Zegarelli Updated2016-03-26 18:31:18From the bookCalculus II For DummiesShare

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An important type of series is called the p-series. A p-series can be either divergent or convergent, depending on its value. It takes the following form:

Here’s a common example of a p-series, when p = 2:

Here are a few other examples of p-series:

Remember not to confuse p-series with geometric series. Here’s the difference:

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

  

• A p-series has the variable in the base — for example

  

As with geometric series, a simple rule exists for determining whether a p-series is convergent or divergent.

A p-series converges when p > 1 and diverges when p

Here are a few important examples of p-series that are either convergent or divergent.

When p = 1: the harmonic series

When p = 1, the p-series takes the following form:

This p-series 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 p-series 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 p-series looks like this:

Because p ≤ 1, this series diverges. To see why it diverges, notice that when n is a square number, say n = k2, the nth term equals

So this p-series includes every term in the harmonic series plus many more terms. Because the harmonic series is divergent, this series is also divergent.

About This Article

This article is from the book: 

Calculus II For Dummies

About the book author:

Mark Zegarelli is a math tutor and author of several books, including Basic Math & Pre-Algebra For Dummies.

This article can be found in the category: 

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