I am working on this sequence $a_n = (1+p)^{p^n}$. I would like to show that it converges in $Q_p$. I am using the property that the sequence converges in $Q_p$ iff it satisfies the following property $\lim_{n\to \infty} |a_{n+1} - a_n|_p = 0$ $$|(1+p)^{p^n}[((1+p)^{p^n})^{p-1} -1]|_p = |((1+p)^{p^n})^{p-1} -1|_p $$ since the term $(1+p)^{p^n}$ is not divisible by $p$ and will not affect the valuation. Now I used binomial theorem and got rid of $1$. I am left with $$|\binom{p^n(p-1)}{1}p +...+p^{p^n(p-1)}|_p$$ now I would like to use the ultrametric property and find the max term. Could someone give me a hint on how to do that?
Thanks.
This sequence converges to $1$ in $\Bbb Q_p$. Let $a_1=1+p$ and $a_{n+1} =a_n^p$. Then we claim $a_n\equiv1\pmod{p^n}$. If true for some $n$, then $a_n=1+bp^n$ and $a_{n+1}=1+bp^{n+1}+$ terms all divisible by $p^{n+1}$.