Let an, n>=1 denote a sequence of positive real numbers, and for each n>=1, define rn to be the unique positive solution to the equation
Given a nonnegative number L, characterize those sequences an
such that lim n-->oo an = L.
I created the following problem while I was playing around with some graphs. Consider graphing the following equations:
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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