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I am trying to do the following question

Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$.

In general in $\mathbb{Q}_p$ what is the stronger condition, to be p-adically convergent or p-adically Cauchy?

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    I did not mean to leave it out!2012-04-26

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I guess the thing to notice is that in the $p$-adics we have a geometric series \[ \frac{1}{1 - p} = 1 + p + p^2 + \cdots; \] you can prove this as before, noting that |p| < 1 in our new absolute value. With $p = 7$ the left side is $-1/6$, so the partial sums of the right side form a sequence in $\mathbf Z$ which converges to that number.

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    Alternatively, if you want to find the $p$-adic expansion of an element of the localization $\mathbf Z_{(p)}$, then you really can just look at the image of that element in $\mathbf Z/p\mathbf Z$, $\mathbf Z/p^2\mathbf Z$, etc. This might be better if the number you want to approximate isn't part of a well-known formula. Let me know if you want me to say more about this.2012-04-26