How to prove a series doesn't converge in $\mathbb{Z}_{p}\langle x \rangle$

Question

Suppose a seires $A=\sum_{i=1}^{\infty}(1+px)^i$ and I was trying to prove it doesn't converge in $\mathbb{Z}_{p}\langle x \rangle$. Is it correct to say that because we know that $|1+px|_{p}=1$, it cannot converge w.r.t the p-adic norm?

Answer 1

Yes, it is, since in $\;p\,-$ adic analysis convergence of series is pretty simple: a series $\;\sum a_n\;$ converges iff $\;a_n\xrightarrow[n\to\infty]{}0\;$, and this isn't fulfilled here by what you wrote.

I never saw the notation $\;\Bbb Z_p\langle x\rangle\;$ , which apparently means ring of power series, which I know by $\;\Bbb Z_p[[x]]\;$ .