p test

The following is an immediate corollary of the integral testMathworldPlanetmath.

Corollary (p-Test).

A series of the form n=11np converges if p>1 and diverges if p1.


The case p=1 is well-known, for n=11n is the harmonic seriesMathworldPlanetmath, which diverges (see this proof (http://planetmath.org/ProofOfDivergenceOfHarmonicSreies)). From now on, we assume p1 (notice that one could also use the integral test to prove the case p=1). In order to apply the integral test, we need to calculate the following improper integral:


Since limnnt diverges when t>0 and converges for t0, the integral above converges for 1-p<0, i.e. for p>1 and diverges for p<1 (and also diverges for p=1). Therefore, the corollary follows by the integral test. ∎

Title p test
Canonical name PTest
Date of creation 2013-03-22 15:08:51
Last modified on 2013-03-22 15:08:51
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Corollary
Classification msc 40A05
Synonym p-test
Synonym p-test
Synonym p test
Synonym p series test
Synonym p-series test
Synonym p series test
Related topic ExamplesUsingComparisonTestWithoutLimit
Related topic ASeriesRelatedToHarmonicSeries