<s>[INST] Solving an unusual equation I need to find a real number $n$ such that $n > 1$ and:
$$
\sum_{k=1}^\infty \frac{2^k}{n^k} = \frac{n-1}{n}
$$
Ideally, I'd find the minimum such $n$ (if more than one exists), but really, any such $n$ will do.
I cannot for the life of me figure out how to solve this equation for $n$!  Are there any strategies I might be missing?
 [/INST] $$\sum_{k=1}^{\infty}\frac{2^k}{n^k}=\left(\frac{2}{n}\right)+\left(\frac{2}{n}\right)^2+\left(\frac{2}{n}\right)^3+\cdots+\left(\frac{2}{n}\right)^\infty$$
Common Ratio is $\frac{2}{n}$ So, This will converge when $|n|>2$.
$$\sum_{k=1}^{\infty}\frac{2^k}{n^k}=\frac{\frac{2}{n}}{1-\frac{2}{n}}$$
$$\sum_{k=1}^{\infty}\frac{2^k}{n^k}=\frac{\frac{2}{n}}{1-\frac{2}{n}}=\frac{2}{n-2}$$
It's all easy then $$\frac{2}{n-2}=\frac{n-1}{n}$$ gives quadratic in n $$n^2-5n+2=0$$
since $n>2$ we get $$n=\frac{5+\sqrt{17}}{2}$$
</s>