GATE IT 2007 | Question: 77

Question

Consider the sequence $\left \langle x_n \right \rangle,\; n \geq 0$ defined by the recurrence relation $x_{n + 1} = c \cdot (x_n)^2 - 2$, where $c > 0$.

For which of the following values of $c$, does there exist a non-empty open interval $(a, b)$ such that the sequence $x_n$ converges for all $x_0$ satisfying $a < x_0 < b$?

1. $0.25$
2. $0.35$
3. $0.45$
4. $0.5$
   • A. i only
   • B. i and ii only
   • C. i, ii and iii only
   • D. i, ii, iii and iv

Comments

Non-gate tag means out of syllabus?

Answer 1

Answer: D

For the series to converge the limit: n tends to infinity of (x_n+1/x_n) should be < 1. 

From the recurrence we should have cx_n² - x_n - 2 < 0.

For all the above values of c we have the above equation as negative.

Comments

For all the above values of c we have the above equation as negative , plz elaborate this..

---

What is the meaning of....xn converges for all x0

---

Series converges to a value .it  is true for all value of x within the interval:a<x<b