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?

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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.

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

HallsofIvy
Homework Helper
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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