The College Mathematics Journal, Vol. 31, No. 5, Nov. 2000, problem 688, p. 409

11/2000

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.

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Hint:

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

1. y = x² + 2x, y = 2
2. y = x³ + 3x² + 6x, y = 6
3. y = x⁴ + 4x³ + 12x² + 24x, y = 24
4. y = x⁵ + 5x⁴ + 20x³ + 60x² + 120x, y = 120

etc. (those familiar with differentation of polynomials will notice the pattern)

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: