What does power series converge p-adically means??

Question

What is P-adically and, what does p-adically converge mean?

I have no idea what it is, really. I've trying to google it however, I failed so badly so I had to bring this question.

Answer 1

Fractions, written in decimals, can have infinitely many digits after the decimal point.
$p$-adic numbers have infinitely many digits before the decimal point instead.
So instead of 3.1415926535.... you have ...25643.4
The reason for $p$ is that division doesn't work properly in base-10. Two non-zero numbers can multiply to give zero, which means that division can give more than one answer. It has to be a prime base instead. So $p$ is for prime.

Now to converge $p$-adically: The sequence $$1,21,321,2321,12321,212321,...$$ converges $p$-adically to the $p$-adic $...212321$.
The difference between normal decimals depends on the left-most place that they differ - so 1.931 and 1.932 are closer than 1.931 and 1.941. The further right the first difference, the closer the numbers are together.
The difference between $p$-adics depends on the right-most place that they differ, so ...1221 and ...1121 are closer than ...1221 and ...1231 The further left the first difference, the closer the $p$-adics are together.
If the right-most difference is in the 10s place, the distance is 1/10; if the difference is in the 100s place, the distance is 1/100.
In base $p$, the distances are $1/p$ and $1/p^2$ respectively.