Let a_{n}, n>=1 denote a sequence of positive real numbers, and for each n>=1, define r_{n} to be the unique positive solution to the equation

Given a nonnegative number L, characterize those sequences a_{n}
such that lim n-->oo a_{n} = L.

Hint:

I created the following problem while I was playing around with some graphs. Consider graphing the following equations:

- y = x
^{2}+ 2x, y = 2 - y = x
^{3}+ 3x^{2}+ 6x, y = 6 - y = x
^{4}+ 4x^{3}+ 12x^{2}+ 24x, y = 24 - y = x
^{5}+ 5x^{4}+ 20x^{3}+ 60x^{2}+ 120x, y = 120

Notice that the x value of the intersection for each pair of equations decreases. However, if the process were continued, we know the x value of the intersection would not be 0, because we'd have

which is not possible. Therefore the x value must be greater than 0, but what x value is it?

Here is the correspondence I have on the problem:

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