Show that $\sum_{n=0}^{\infty}p^n$ convergence in $\mathbb{Z_p}$

Question

For a prime $p$ , I need to show that $$\sum_{n=0}^{\infty}p^n$$ converges in $\mathbb{Z_p}$ (p-adic integers) .

Now since $$\sum_{n=0}^{\infty}p^n = \frac{1}{1-p} \in \mathbb{Z_p} $$ so I suppose one way to show the convergence is to show that this is a Cauchy sequence. But I am not able to do so.

Any suggestions!

Answer 1

The valuation of the distance of the $n$-th partial to the limit is $n+1$, as the difference is $\frac{p^{n+1}}{p-1}$. This already proves convergence in the $p$-adic metric.