p test
The following is an immediate corollary of the integral test.
Corollary (p-Test).
A series of the form ∑∞n=11np converges if p>1 and diverges if p≤1.
Proof.
The case p=1 is well-known, for ∑∞n=11n is the harmonic series, which diverges (see this proof (http://planetmath.org/ProofOfDivergenceOfHarmonicSreies)). From now on, we assume p≠1 (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:
∫∞11xp𝑑x=limn→∞[x1-p1-p]n1=limn→∞n-p+11-p-11-p. |
Since limn→∞nt diverges when t>0 and converges for t≤0, 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 |