Question

The convergence of the following series
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Answer 1

Step-by-step explanation:

have to study the convergence of the following series:

∑n≥1n!p(p+1)⋯(p+n−1) where p>0.

I tried d'Alembert criterion but limn→∞an+1an=1 (where an=n!p(p+1)⋯(p+n−1)).

Because that limit is 1 the nature of the series is inconclusive.

Intuitively I can say that the series is convergent because when p∈{1,2,n} the sum becomes:

For p=1, ∑n≥1n!1⋅2⋯(1+n−1)=∑n≥1n!n!=n is convergent

and

For p=2, ∑n≥1n!2⋅3⋯(2+n−1)=∑n≥1n!2⋅3⋯(n+1)=∑n≥1n!(n+1)!=∑n≥11n+1 is convergent

and

For p=n, ∑n≥1n!n(n+1)⋯(2n−1)=∑n≥1(n−2)!(n+2)(n+3)⋯(2n−1) is convergent (d'Alembert)