P-adic convergence

  • Thread starter Oxymoron
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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?
 
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.
 
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]
 

matt grime

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why do'nt you just work out the partial sums? it is a geometric series.
 

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?
 
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.
 
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