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P-adic convergence

  1. Aug 18, 2005 #1
    Question

    Prove that the series [itex]\sum_{n=0}^{\infty} p^n[/itex] converges in the p-adic metric by showing that the sequence of partial sums converge. What does the series converge to?
     
  2. jcsd
  3. Aug 18, 2005 #2
    Solution

    Let [itex]s_m = \sum_{n=0}^m p^n[/itex] be the sequence of partial sums. Then

    [tex]|s_{n+1} - s_n|_p = |p^{n+1}|_p[/tex]

    Now

    [tex]|p^{n+1}|_p = \frac{1}{p^{n+1}} \rightarrow 0[/tex] as [itex]m,n \rightarrow \infty[/itex] independently in [itex]\mathbb{R}_p[/itex].

    Hence the sequence of partial sums [itex]s_m[/itex] converges and the series converges to 0.
     
  4. Aug 18, 2005 #3
    Does this solution look correct to anyone?

    Also, I think that the sequence is Cauchy since

    [tex]\lim_{n\rightarrow \infty}^p |p^{n+1}|_p = 0[/tex]
     
  5. Aug 19, 2005 #4

    matt grime

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    why do'nt you just work out the partial sums? it is a geometric series.
     
  6. Aug 19, 2005 #5

    HallsofIvy

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    This is the third post in which you've immediately answered your own question. What is your purpose in posting them?
     
  7. Aug 19, 2005 #6
    Hey Halls,

    I have no idea that my solutions are correct! If they are...that's great!

    The sticky on this forums says not to expect any help unless you have a go at the problem first yourself...so I do. If there is nothing wrong with them, please by all means, tell me so I know.
     
    Last edited: Aug 19, 2005
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