P-Series: Proof, Test, Integral & Convergence

This article will explore a generalization of the harmonic series, p-series, its definition or sum, criterion of convergence, and other related convergence tests.p-Series SumSo first of all, what is a harmonic series? It is a kind of p-series. Now you need to know what a p-series is. A p-series is a series of the form ∑n=1∞1np = ∑n=1∞1np where p is a real number.When this is called the Harmonic series.Decide if the series ∑n=1∞n4n5 is a p-series or not.Answer:It doesn't look like a p-series at first glance, but let's do some algebra just to be sure. Heren4n5 = n14n5 = 1n5·n-14 = 1n5-14 = 1n194 = 1n194 so in fact this is a p-series.Is the series ∑n=1∞13n a p-series?Answer:No, because in a p-series it needs to be 1n raised to a constant power, not a constant raised to thenpower. In fact this series also has a special name, it is called a Geometric series. For more information on this type of series see Geometric Series.p-Series and the Integral TestFor more information on why the Integra...