Why is partially summing terms of an infinite p-adic series unjustified? Why is this unjustified? $$\sum_{n=1}^\infty 1$$ $$= \sum_{n=1}^p 1 +\sum_{n=p+1}^{p+p^2} 1 + \sum_{n=p+p^2+1}^{p+p^2+p^3} 1 + \cdots$$ $$=p+p^2+p^3+\cdots$$ $$ = \frac{p}{1-p}$$

Whereas the second "=" is true in general (if "+..." is interpreted correctly), and the third "=" is true with respect to the $p$-adic metric, the first "=" lacks justification. It would be some "infinite associativity law" which simply does not exist if the series is not convergent.

To see things clearer, remember that convergence of a series $\sum a_n$by definition means convergence of the sequence of partial sums $s_n = \sum_{1}^n a_i$. Now in this example, $s_n =n$ which obviously does not converge in any $p$-adic metric. However you cleverly spotted a subsequence $s_p, s_{p+p^2}, s_{p+p^2+p^3}, ...$ which converges $p$-adically to $\frac{p}{p-1}$. Now _if_ the sequence $s_n$ did converge, then it would converge to the same limit, but _it does not_ \-- it just happens to have this neat convergent subsequence.