# Divergence in p-adic metric

Why does the series of 1/n! diverge in the p-adic metric?In other words, how do I show that the lim of 1/n! (in the p-adic metric) does not equal 0 because it is >1

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## Answers and Replies

Hurkyl
Staff Emeritus
Gold Member
Seems obvious to me -- can you find a lower bound on how many times p goes into n!?

No, but I am not sure how to show this. I know that every n has a prime factorization, and that as n increases, p will go into n more and more times. But what does this mean?

matt grime
Homework Helper
COnsider the subsequence 1/p!, 1/(p^2)!, 1/(p^3)! What is the p-adic valuation of 1/(p^r)! at least as great as?

The only answer to that question that makes sense to me is 1

Hurkyl
Staff Emeritus
Gold Member
No, but I am not sure how to show this. I know that every n has a prime factorization, and that as n increases, p will go into n more and more times. But what does this mean?

So, how does the p-adic valuation of n! relate to the number of times p goes into n!?

How does that relate to the p-adic valuation of 1/(n!)?

matt grime