18 votes

Consider the sequence $\langle x_n \rangle , \: n \geq 0$ defined by the recurrence relation $x_{n+1} = c . x^2_n -2$, where $c > 0$.

Suppose there exists a ** non-empty, open** interval $(a, b)$ such that for all $x_0$ satisfying $a < x_0 < b$, the sequence converges to a limit. The sequence converges to the value?

- $\frac{1+\sqrt{1+8c}}{2c}$
- $\frac{1-\sqrt{1+8c}}{2c}$
- $2$
- $\frac{2}{2c-1}$

10

I am getting option B using hit and trial.

Let

Then,

So, the value converges to , which is equal to

Which means that B is correct.

3

Convergence, inmathematics, property (exhibited by certain infinite series and functions) of approaching a limit more and more closely as an argument (variable) of`or as the number of terms of the series increases. For example, the function y = 1/x converges to zero as x increases.`

`the function increases or decreases`

14 votes

Best answer

Lets take a look when $c=1$

The value the recurrence converges to must be, $\dfrac{1\pm\sqrt{1+8\times1}}{2\times 1} =2,{-1} $

However, when we take the positive square root, i.e. when $x_{i}$ supposedly converges to $\dfrac{1+\sqrt{1+8c}}{2c}$,

the convergence does not hold for the neighborhood.

$\quad x_{i}=\lim\limits_{\delta \to 0} 2+\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0}\left(2+\delta\right)^{2}-2$

$\qquad=\lim\limits_{\delta \to 0} 4-2+\delta ^{2}+4\delta$

$x_{i+1}=\lim\limits_{\delta \to 0} 2+\delta ^{2}+4\delta$

We can see that $x_{i+1}$ is further than $x_{i}$ from the assumed convergence value of $\lim\limits_{i \to \infty} x_{i}=2$

Similarly, the value does not converge when $x_{i}$ approaches $2$ from the left side of the number line.

When the negative square root is considered, the convergence does hold for neighbors on either side.

$\quad x_{i}=\lim\limits_{\delta \to 0} \left({-1}\right)+\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0} \left(\left({-1}\right) +\delta\right)^{2}-2$

$\qquad =\lim\limits_{\delta \to 0} 1-2+\delta ^{2}-2 \delta$

$x_{i+1}=\lim\limits_{\delta \to 0} -1+\delta^{2}-2\delta$

Also,

$\quad x_{i}=\lim\limits_{\delta \to 0} \left({-1}\right)-\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0} \left(\left({-1}\right) -\delta\right)^{2}-2$

$\qquad =\lim\limits_{\delta \to 0} 1-2+\delta ^{2}+2 \delta$

$x_{i+1}=\lim\limits_{\delta \to 0} -1+\delta^{2}+2\delta$

Hence, when negative square root is considered, the value oscillates around the convergence point, and actually converges.

Therefore the answer should be only B.

The value the recurrence converges to must be, $\dfrac{1\pm\sqrt{1+8\times1}}{2\times 1} =2,{-1} $

However, when we take the positive square root, i.e. when $x_{i}$ supposedly converges to $\dfrac{1+\sqrt{1+8c}}{2c}$,

the convergence does not hold for the neighborhood.

$\quad x_{i}=\lim\limits_{\delta \to 0} 2+\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0}\left(2+\delta\right)^{2}-2$

$\qquad=\lim\limits_{\delta \to 0} 4-2+\delta ^{2}+4\delta$

$x_{i+1}=\lim\limits_{\delta \to 0} 2+\delta ^{2}+4\delta$

We can see that $x_{i+1}$ is further than $x_{i}$ from the assumed convergence value of $\lim\limits_{i \to \infty} x_{i}=2$

Similarly, the value does not converge when $x_{i}$ approaches $2$ from the left side of the number line.

When the negative square root is considered, the convergence does hold for neighbors on either side.

$\quad x_{i}=\lim\limits_{\delta \to 0} \left({-1}\right)+\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0} \left(\left({-1}\right) +\delta\right)^{2}-2$

$\qquad =\lim\limits_{\delta \to 0} 1-2+\delta ^{2}-2 \delta$

$x_{i+1}=\lim\limits_{\delta \to 0} -1+\delta^{2}-2\delta$

Also,

$\quad x_{i}=\lim\limits_{\delta \to 0} \left({-1}\right)-\delta$

$x_{i+1}=1\times x^{2}_{i}-2=\lim\limits_{\delta \to 0} \left(\left({-1}\right) -\delta\right)^{2}-2$

$\qquad =\lim\limits_{\delta \to 0} 1-2+\delta ^{2}+2 \delta$

$x_{i+1}=\lim\limits_{\delta \to 0} -1+\delta^{2}+2\delta$

Hence, when negative square root is considered, the value oscillates around the convergence point, and actually converges.

Therefore the answer should be only B.

0

When you're going for higher order terms such as etc with the condition the values are actually diminishing. and so on. And all these values become 0. What you may want to look at maybe is or .

Secondly, convergence can happen from one side only. For converging to a value, lets say , why is it necessary that the iteration should converge for and where at the same time? Only one of them can hold true for convergence.

0

Ignore the higher order terms, and notice that when ,

For example when

So, the value does not converge.

I had to show that the value does not converge from either side when the positive square root is taken, hence I proved them both.

19 votes

A sequence converges means at some point $x_{n+1} = x_{n}$

Then,

$x=cx^2 -2$

or

$cx^2 -x -2 = 0$

Solving for $x$:

$x=\frac{1\pm \sqrt{1+8c}}{2c}$

So both (**A**) and (**B**) can be the values.

10 votes

Here,

$x_{n+1} = cx_{n}^{2}-2 , c > 0$

For stability, We can write non-linear first order recurrence as :- $x_{n} = f(x_{n-1})$ (or) $x_{n+1} = f(x_{n})$

So, $x_{n+1} = cx_{n}^{2}-2 , c > 0$ becomes $x_{n} = cx_{n}^{2}-2 , c > 0$ (or) For simplicity, to find fixed stable points , we can write it as :- $x = cx^{2}-2 , c > 0$

Now, after solving the given equation , we will get :-

$x= \frac{1\pm \sqrt{1+8c}}{2c}$

Now, here we have $2$ fixed points $x_{1}= \frac{1 + \sqrt{1+8c}}{2c}$ and $x_{2}= \frac{1 - \sqrt{1+8c}}{2c}$

Now, We have to check that for which fixed point the given recurrence converges.

As we know if $f' < 0$ at a point or slope is negative then it means function $f$ is strictly decreasing and if $f' > 0$ then it means function $f$ is strictly increasing.

So, if we find $f' < 0$ in a given interval , it means function $f$ is decreasing or converses to a fixed point in the given interval.

So, Now, Since here , $f(x) = cx^{2} - x -2$

So, $f'(x) = 2cx - 1$

Now,

$f'(x_{1}) = 2c * \left ( \frac{1+\sqrt{1+8c}}{2c} \right ) - 1$

So, $f'(x_{1}) \nless 0 , c> 0$

Now,

$f'(x_{2}) = 2c * \left ( \frac{1-\sqrt{1+8c}}{2c} \right ) - 1$

So, $f'(x_{2}) < 0 , c> 0$

So, here local stable point is $\frac{1-\sqrt{1+8c}}{2c}$ for which given recurrence converges.

So, Answer is **(B)**

Reference :- http://www.ms.uky.edu/~droyster/ma114F16/RecursiveSequences.pdf