I don't think I understand how p-adic numbers relate to the usual concept of infinity. The wiki page and various sources on the internet did not help.
Let's see for example the 10-adic counterparts to the real numbers (10 is not prime, but it is more convenient here for demonstration purposes):
$$ 0 \rightarrow 0 \\ 1 \rightarrow 1 \\ 2 \rightarrow 2 \\ $$
Anyway, for any positive whole number we have the usual representation. On the other hand:
$$ -1 \rightarrow ...9999 \\ -2 \rightarrow ...9998 \\ -3 \rightarrow ...9997 \\ $$
But if $...9999$ is not equivalent to $+ \infty$ then where is the infinity? Is zero the infinity?
$$ -1+1=0 \rightarrow ...0000=0 \\ $$
Do negative and positive infinities 'meet in the middle' between $0$ and $-1$?
It also seems that many infinite series divergent in the field of reals, converge in the field of p-adic numbers. This question is about the divergent series, but I am more interested in the general concept of infinity for p-adics.
For now it seems to me that p-adics incorporate everything in the same form - whole numbers, rational and irrational numbers, negative numbers and even infinity.