something I just thought of is that maybe I can apply a version of the ratio test with the p-adic absolute value in place of the normal absolute value?
So that way the p-adic abs. value of (p^k+1)/(p^k) = p-adic abs. of p which equals to 1/p [by definition of p-adic abs. value]. So this being less than 1, can I conclude that the sequence converges?