Why does the series 1/n! diverge in the p-adic metric?

[Ed Quanta]

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

 

[Hurkyl]

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

 

[Ed Quanta]

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]

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?

 

[Ed Quanta]

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

 

[Hurkyl]

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]

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

Eh? (p^r)! how many times at least must p divide this? You can do better than 1, surely? find a multiple of 2 dividing 4! such as 4, one for 8! such as 8, what about 16!?