<s>[INST] 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?
 [/INST] This can solved from scratch by taking one term out of the summation, changing the index range and refactoring to obtain
$$
\begin{align}
S=
\sum_{i=0}^{\infty} \left(\frac{2}{x}\right)^i
&=
1+\sum_{i=1}^{\infty} \left(\frac{2}{x}\right)^i \\
&=
1+\sum_{i=0}^{\infty} \left(\frac{2}{x}\right)^{i+1} \\
&=
1+\frac{2}{x}\sum_{i=0}^{\infty} \left(\frac{2}{x}\right)^{i} \\
\end{align}$$
Now grouping the series terms yields
$$
\begin{align}
\left(1-\frac{2}{x}\right)\sum_{i=0}^{\infty} \left(\frac{2}{x}\right)^i
&=
1 \\
\sum_{i=0}^{\infty} \left(\frac{2}{x}\right)^i
&=\frac{1}{1-\frac{2}{x}}=\frac{x}{x-2} \\
\end{align}$$
where the sum converges for $\vert \frac{2}{x} \vert\lt 1$, not only $x \ge 3$.
Are you assuming positive integer $x$?
</s>