Consider the sequence <xₙ>, n ≥ 0, defined by the recurrence relation xₙ₊₁ = c…
2007
Consider the sequence <xₙ>, n ≥ 0, defined by the recurrence relation
xₙ₊₁ = c xₙ² - 2, where c > 0.
Suppose there exists a non-empty open interval (a, b) such that for all x₀ satisfying a < x₀ < b, the sequence converges to a limit. The sequence converges to the value
- A.
(1 + √(1 + 8c))/(2c)
- B.
(1 - √(1 + 8c))/(2c)
- C.
2
- D.
2/(2c - 1)
Show answer & explanation
Correct answer: B
If the sequence converges to a limit L, then L must satisfy the fixed-point equation
L = cL² - 2.
So, cL² - L - 2 = 0. Solving this quadratic gives
L = (1 ± √(1 + 8c))/(2c).
The positive root is (1 + √(1 + 8c))/(2c). For f(x) = cx² - 2, we have f'(x) = 2cx. At the positive root, |f'(L)| = 1 + √(1 + 8c) > 1, so it is repelling and cannot be the limit for all initial values in a non-empty open interval.
The attracting fixed point, when such an interval of convergence exists, is therefore the negative root:
L = (1 - √(1 + 8c))/(2c).