Convergence of Infinite Sum Using Mathematica, I have been able to make the following statement based on numerical evidence:
$$\sum_{i=0}^\infty \frac{2^i}{x^i}=\frac{x}{x-2}$$
for any $x≥3$. How can this be proven?

Define $r = 2/x$. Then
$$\begin{align*}
\sum_{i=0}^\infty \frac{2^i}{x^i} = \sum_{i=0}^\infty r^i = \frac{1}{1-r} = \frac{x}{x-2}
\end{align*}$$
where the summation formula for geometric series is used.
