# The $p$-adic expansion of a function of $p$

Let $p\neq 2$ be prime. I am asked in a revision question to find the $p$-adic expansion of $(1+2p)/(p-p^3)$. The best I could do was find the $p$-adic norm, which I got as $p$ (please correct me if this is wrong) thus giving the expansion as $$\frac{1+2p}{p-p^3} = a_{-1}p^{-1} + a_0 + a_1 p + \dots$$ where $a_i \in \{0,\dots,p-1\}$. I've tried several other things and none of them seem to be fruitful. No questions of this type were even addressed in lectures... Can anyone help?

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Write it as $$\frac{1+2p}{p}\cdot\frac{1}{1-p^2}.$$ Because $|p^2|_p=\frac{1}{p^2}<1$, we can use the geometric series to give the $p$-adic expansion of $\frac{1}{1-p^2}$. Then shift it down by one due to the $\frac{1}{p}$, then multiply by $1+2p$ (the action of this on the coefficients is not too hard to figure out).