1. The problem statement, all variables and given/known data

Let p be a prime number. Which of the following series converge p-adically? Justify your answers: (all sums are from n = 0 to infinity)

(i) Ʃp^n

(ii) Ʃp^-n

(iii) Ʃn!

(iv) Ʃ (2n)! / n!

(v) Ʃ (2n)! / (n!)^2

2. Relevant equations

The definition given for p-adic convergence is the following:

Ʃx_n converges p-adically iff U_p(x_n) → ∞

Here we define U_p (n) = max { a : p^a | n }

3. The attempt at a solution

1. is easy since U_p (p^n) = nU_p(p) = n -> inf

2. if it tends to -inf then is it p-adically convergent based on the above?

3-5. I am completely lost. To be honest I really just need some examples, but I am barely able to find anything regarding this subject on the internet. Why is that? Can anyone explain how to do (iii) for instance?

The lecturer explained how to find the identity U_p (n!) ≤ n/(p-1) but that doesn't really help me here, if x is less than y and y -> infinity, this does not imply x-> infinity.