I was trying to find the solution to the formula:
$$x = \sum_{n=1}^\infty{x^{-n}}$$
I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$.
I do not know if this has already been found, but I thought that it was a cool discovery.
My question is, has this already been found, and, if it has, what implications, or uses, does it have in the world of Mathematics, if any?
Something like it is used in John Conway's analysis of a puzzle about peg solitaire. To quote from the Wikipedia essay,
A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible.
Conway proved that, regardless of the strategy used, there is no finite series of moves that will allow a soldier to advance more than four rows above the horizontal line. His argument uses a carefully chosen weighting of cells (involving the golden ratio), and he proved that the total weight can only decrease or remain constant.
For the details, see the essay. It relies on $$\sum_{n=2}^{\infty}\phi^n=1$$ where $\phi=(\sqrt5-1)/2$.