Sequence in Q with p-diatic metric. Show it converges to a rational
This is the problem I'm trying to slove:
Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime).
Show that s_n converges to a rational number.
Now, I do get some intuition on showing that the number it converges to is rational by applying the fundamental theorem of arithmetic and claiming that the number it converges to can be expressed as a product of primes so that way we can factor out p to some power and satisfy the convergence definition with the p-diatic metric.
My big problem is: How can I show that s_n converges in the first place? or should i start my proof by assuming that it converges to some s, and then showing that s is rational?
Re: Sequence in Q with p-diatic metric. Show it converges to a rational
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?