Divergence in p-adic metric

  • Thread starter Ed Quanta
  • Start date
  • #1
297
0
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
 
Last edited:

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
Seems obvious to me -- can you find a lower bound on how many times p goes into n!?
 
  • #3
297
0
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?
 
  • #4
matt grime
Science Advisor
Homework Helper
9,395
4
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?
 
  • #5
297
0
The only answer to that question that makes sense to me is 1
 
  • #6
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
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!)?
 
  • #7
matt grime
Science Advisor
Homework Helper
9,395
4
Ed Quanta said:
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!?
 

Related Threads on Divergence in p-adic metric

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
1
Views
4K
  • Last Post
Replies
8
Views
792
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
3
Views
696
Replies
6
Views
3K
Replies
3
Views
3K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
5
Views
1K
Top