Studying $p$-adic numbers I encountered the following theorem:
Given a eventually periodic sequence $(a_n)_{n=k}^{\infty}$ such that $0 \le a_n <p$, the sum
\begin{equation*}
\sum_{n=k}^{\infty}a_np^n
\end{equation*}
converges p-adically to a rational number.
The proof of this fact consists mainly in rearranging the sum. Here is my problem… In all the books I have seen this is not justified. Only some authors prove a theorem about rearrangement, but in other parts of their books and seems we don't need it here.
Why I can rearrange the terms of this sum? Why I don't need any theorem?
Thank you all!
Best Answer
As @ThomasAndrews says, you don’t need to rearrange anything. First, knock off the nonperiodic part at the beginning. That will be a rational number. Now take the periodic part, say of period $N$. Then the part you didn’t knock off has the form $A + Ap^N + Ap^{2N} +Ap^{3N}+\cdots$, a geometric series with common ratio $p^N$. Since this ratio is $p$-adically smaller than $1$, the series is convergent, and the periodic part has rational value $A/(1-p^N)$, done.