**If you find any of this useful, please consider donating via PayPal to help keep this site going.**

**Email news@statisticool.com to sign up to receive news and updates**

__The College Mathematics Journal__, Vol. 31, No. 5, Nov. 2000, problem 688, p. 409

**11/2000**

_{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

0 = really large number

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:

_{}

If you enjoyed *any* of my content, please consider supporting it in a variety of ways:

- Email news@statisticool.com to sign up to receive news and updates
- Donate any amount via PayPal
- Take my Five Poem Challenge
- Buy ad space on Statisticool.com
- Visit my Amazon author page
- Buy what you need on Amazon using my affiliate link
- Follow me on Twitter here