How is the ternary expansion of $1=0.222...$?

Question

I understand how the ternary expansion of $1=1.00$... but I don't understand how $1=0.22$... .

Answer 1

In ternary:

$\begin{eqnarray}0.222222... & = & \sum\limits_{i=1}^{\infty}2\frac 1 {3^i}\\ & = & \lim\limits_{n\to \infty} \sum\limits_{i=1}^{n}2\frac 1 {3^i}\\ & = & \lim\limits_{n\to \infty}1 -\frac 1 {3^{n}}\\ & = & 1.\end{eqnarray}$

This is the exact same reason why $0.99999...=1$ in base $10.$