Solve $\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+\cdots\right)$ If
$$\displaystyle \frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$
then what is $x$?
I was thinking of expanding the brackets and trying to notice a pattern but as it effectively goes to infinity. I don't think I can expand it properly, can I?
 A: There is an obvious symmetry to this problem. If infinite sum converges then notice that
$$\frac{1}{2x} + \frac{1}{2}\left(y\right) = y$$
Rearranging and solving gives $\frac{1}{x}=y$ and so $x=\frac{1}{y}$.
A: Hint Note that what is inside the outer pair of parenthesis is equal to $y$.
A: Expand. The first term is $\frac{1}{2x}$.
The sum of the first two terms is $\frac{1}{2x}+\frac{1}{4x}$.
The sum of the first three terms is $\frac{1}{2x}+\frac{1}{4x}+\frac{1}{8x}$.
And so on.
The sum of the first $n$ terms is 
$$\frac{1}{2x}\left(1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n-1}}\right).$$
As $n\to\infty$, the inner sum approaches $2$. So the whole thing approaches $\frac{1}{x}$.
So we are solving the equation $\frac{1}{x}=y$.
A: Think about how pretty it is to note that $$\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$ implies $$\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = 2y-\frac{1}{x}=y.$$ Now, solve for $y$. Of course, I'm assuming we don't need to care about convergence here at all.
