Convergence in $\mathbb{Q}_{p}$.

Question

I do not know how to show that:

$1+p+p^{2}+\ldots$ converges to $\frac{1}{1-p}$ in $\mathbb{Q}_{p}$ where $\mathbb{Q}_{p}$ is the set of $p$-adic numbers.

Could someone help me?

Answer 1

Suggestion/hint: Write $s_n = 1+p +\cdots + p^n$. Then $ (1-p) s_n -1 = -p^{n+1} $. Divide throughout by $(1-p)$: $$ s_n - {1\over 1- p} = {-p^{n + 1} \over 1- p }. $$ The absolute value of the RHS tends to zero as $n\to\infty$.

Answer 2

The difference between them is a sum of very high powers of $p$. In other words, it is very small in the $p$-adic metric.