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If we have the following $p$-adic number:

$$2+3p+5p^2+2p^3+3p^4+5p^5+2p^6+3p^7+5p^8+.....$$ and I am trying to find what rational number this p-adic number represents. I have no idea as to how to go about doing this. The only thing that I can notice is that maybe we need to split this up into cases where $p<5$ and $p\geq 5$ because otherwise this is some non-standard representation.

Cheers

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    $\begingroup$ group the terms in threes, and note that each group is $p^3$ times the preceding group. So you have a geometric series, with ratio $|p^3|\,<\,1$. From here on, you do it just as you did in high school. $\endgroup$
    – Lubin
    Apr 27, 2013 at 17:56
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    $\begingroup$ multiply it by $p^3-1$ and tell us what happens. $\endgroup$
    – mercio
    Apr 27, 2013 at 18:13
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    $\begingroup$ thanks, if we sum that geometric sum we get $-\frac{(2+3p+5p^2)}{p^3-1}$. $\endgroup$
    – user73957
    Apr 27, 2013 at 19:08

1 Answer 1

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This is a geometric progression, $$Sum = (2+3p+5p^2)(1+p^3+p^6+...)=\frac{2+3p+5p^2}{1-p^3}$$

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