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Divergence in p-adic metric

  1. Apr 12, 2005 #1
    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: Apr 12, 2005
  2. jcsd
  3. Apr 12, 2005 #2

    Hurkyl

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    Seems obvious to me -- can you find a lower bound on how many times p goes into n!?
     
  4. Apr 13, 2005 #3
    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?
     
  5. Apr 13, 2005 #4

    matt grime

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    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?
     
  6. Apr 13, 2005 #5
    The only answer to that question that makes sense to me is 1
     
  7. Apr 13, 2005 #6

    Hurkyl

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    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!)?
     
  8. Apr 14, 2005 #7

    matt grime

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    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!?
     
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