Hello. I'm having problem with these:
a). Prove that the Bisection algorithm gives a sequence that has the bound of error which converges linearly to zero.
b). The sequence described as , , and , is called Fibonacci sequence and exists, prove that that limit is . (This number is called golden ratio)
Thank you for your time.
Now we are told that the limit as x \to \infty exists, so the limit must satisfy the equation:
which may be simplified to:
which has roots:
Of these roots we may discard the one with the negative sign as must be greater than (in fact greater than ), so the required root is: