P-adic integers Show that $\frac{2}{p-1}$ is a $p$-adic integer and find its p-adic expansion. P-adic numbers really make little sense to me so any help explaining what to do and why would be really appreciated. Thanks

If $p=2$ then your expression is just equal to $2\in\Bbb Z$.

If $p$ is odd then numerator and denominator are prime to $p$, thus $p$-adic units. The quotient of $p$-adic units is a $p$-adic unit. Recall that a $p$-adic unit is an invertible element in the ring $\Bbb Z_p$ and can be recognized as those $p$-adic integers having a non-zero "costant" term in their $p$-adic expansion.

In order to get the $p$-adic expansion of $\frac2{p-1}$ just rewrite it as $-2\frac1{1-p}$ and apply Chris Eagle's hint.