7-adic expansion of a rational number

Question

I know that every rational number has a unique 7-adic expansion, now I need help proving that $7/36=\sum\limits_{i=0}^nn7^n$ as a 7-adic integer. I tried using properties of this fraction, like adding it various times, but this didn´t seem to help. I don´t know what else to do. Any help would be great.

Answer 1

The easiest way is to multiply by $-6=1-7$ to get $7+7^2+7^3+7^4+\cdots=7/(1-7)$, which proves it.

For a more abstract argument, Alain Remillard’s idea in his comment is on the mark. When you expand $t/(t-1)^2$ as a series, you get $\sum_{n\ge1}nt^n$. But the evaluation $t\mapsto7$, as a function $\Bbb Z[[t]]\to\Bbb Z_7$ is continuous, ’cause you’re evaluating $t$ to something whose powers go to zero.