Proving p-adic Convergence: Find Series' Limit

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In summary, the conversation is about proving the convergence of the series \sum_{n=0}^{\infty} p^n in the p-adic metric by showing the convergence of the sequence of partial sums. The series converges to 0 and the sequence is also Cauchy. The person posting multiple solutions is unsure if they are correct and is seeking confirmation.
  • #1
Oxymoron
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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?
 
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  • #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.
 
  • #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]
 
  • #4
why do'nt you just work out the partial sums? it is a geometric series.
 
  • #5
This is the third post in which you've immediately answered your own question. What is your purpose in posting them?
 
  • #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.
 
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1. What is p-adic convergence?

P-adic convergence is a concept in mathematics that refers to the convergence of a sequence of numbers in a p-adic metric space. This is a way of measuring distance between numbers that is based on their prime factorizations.

2. How is p-adic convergence different from standard convergence?

P-adic convergence is different from standard convergence in that it takes into account the prime factorizations of numbers, rather than just their magnitude. This allows for a different way of measuring distance between numbers and determining whether a sequence is converging.

3. What is the significance of proving p-adic convergence?

Proving p-adic convergence is important in many areas of mathematics, particularly in number theory and algebra. It allows for a deeper understanding of the properties of numbers and their relationships, and can be used to solve complex equations and problems.

4. What are some common methods for proving p-adic convergence?

There are several methods that can be used to prove p-adic convergence, including the p-adic metric method, the Cauchy criterion, and the Hensel's lemma. Each method involves different techniques and approaches, but ultimately they all aim to show that a sequence of numbers is converging in a p-adic metric space.

5. How is the limit of a p-adic series determined?

The limit of a p-adic series can be determined using various techniques, including the p-adic metric method, the Cauchy criterion, and the Hensel's lemma. These methods involve evaluating the terms of the series and determining whether they meet certain criteria for convergence. If they do, then the limit of the series can be calculated.

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