Proving p-adic Convergence: Find Series' Limit

[Oxymoron]

Question

Prove that the series \sum_{n=0}^{\infty} p^n converges in the p-adic metric by showing that the sequence of partial sums converge. What does the series converge to?

 

[Oxymoron]

Solution

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

|s_{n+1} - s_n|_p = |p^{n+1}|_p

Now

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

Hence the sequence of partial sums s_m converges and the series converges to 0.

 

[Oxymoron]

Does this solution look correct to anyone?

Also, I think that the sequence is Cauchy since

\lim_{n\rightarrow \infty}^p |p^{n+1}|_p = 0

 

[matt grime]

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

 

[HallsofIvy]

This is the third post in which you've immediately answered your own question. What is your purpose in posting them?

 

[Oxymoron]

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.