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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. $0.5$
1. i only
2. i and ii only
3. i, ii and iii only
4. i, ii, iii and iv

edited | 576 views
+1
Non-gate tag means out of syllabus?

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

From the recurrence we should have cxn2 - xn - 2 < 0.

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

by Boss (33.8k points)
0

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

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What is the meaning of....xn converges for all x0
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Series converges to a value .it  is true for all value of x within the interval:a<x<b