For c>0, consider the following quadratic equation:
- - =0 , x>0
Define the sequence { } recursively fixing > 0 and then, if n is an index for which { } has been defined, defining
= sqrt{c+x_n}
Prove that the sequence { } converges monotonically to the solution of the above equation!
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I tried to attempt this problem by setting = 1, and creating a sequence. It turns out to be a monotonic incereasing sequence. But how do we show that the sequence converges to the solution of the quadratic equation?
Also I am unclear about the starting value of . Is it ok to start with = 1, or does the starting value have to be another number?
The difference equation that generates the sequence is...
(1)
... and its solution depends from and . The 'fixed point' [if any] are the solution of the equation...
(2)
... that is, in the case we consider the positive value of the 'square root' ...
(3)
From (3) we deduce that exists only if is and from (2) that in that case any will produce a sequence converging at . The function is represented in the figure for ...
The point is an 'actratting fixed point' and, because the 'slope' of is negative and in absolute value less than 1 [see red line...] for any the generated sequence will converge to without oscillations...
Kind regards