Why log on $1 +p \Bbb{Z}_p$ ($log:1 +p \Bbb{Z}_p→\Bbb{C}_p)$is continuous ? We can define p-adic logarithm on $x∈1 +p \Bbb{Z}_p$ by $log(1＋x)＝x-x^2/2＋x^3/3-x^4/4・・・$.
My book reads, log is clearly continuous.
Why can I say log is continuous ? I know convergence radius is $1$. All topology is p adic topology.