<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] We can begin by considering the series:
$$\frac{1}{1-x} = \sum_{k=0}^\infty{ x^k } \tag{1}$$
$$\frac{1}{1-x} - 1 = \sum_{k=1}^\infty{ x^k } \tag{2}$$
Then let $x=\frac{2}{n}$:
$$\frac{1}{1-\left(\frac{2}{n}\right)} - 1 = \sum_{k=1}^\infty{ \left(\frac{2}{n}\right)^k } \tag{3}$$
Now we have the desired sum, and we want this equal to $\frac{n-1}{n}$:
$$\frac{1}{1-\left(\frac{2}{n}\right)} - 1 = \frac{n-1}{n} \tag{4}$$
You can probably guess how to proceed from here.  Wolfram Alpha gives these answers:
Solving for $n$, we get:
$$n=\begin{cases}
\frac{5+\sqrt{17}}{2} \\
\frac{5-\sqrt{17}}{2}
\end{cases} \tag{5}$$
Of course, the greater value is $\frac{5+\sqrt{17}}{2}$.
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