What dose {x_n} converges ? when x_1=5/2,5x_(n+1)=x_n^2+6

1 Answer
Aug 7, 2017

See below.

Explanation:

This is a nonlinear difference equation. Normally it is hard to analyze convergence in this case. So we will make some basic convergence considerations.

If x_n converges to x^@ then

5x^@=(x^@)^2+6 and solving for x^@ we have

x^@ = {2,3}

Checking for x = 2+delta we conclude that x^@=2 is an stable attraction point for -oo < delta < 1 and x^@ = 3 is an inestable attraction point. Resuming, for -oo < x_1 < 3 the sequence converges to 2 otherwise is diverges.