- #1

gikiian

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## Homework Statement

Show that the sequence [itex]{(p_{n})}^{∞}_{n=0}=10^{-2^{n}}[/itex] converges quadratically to 0.

## Homework Equations

[itex]\stackrel{limit}{_{n→∞}}\frac{|p_{n+1}-p|}{|p_{n}-p|^{α}}=λ[/itex]

where

- α is order of convergence; α=1 implies linear convergence, α=2 implies quadratic convergence, and so on, provided that 0<λ<1.
- λ is the asymptotic error constant; in scope of the course (Numerical Analysis), 0<λ<1, so I do not have to play with my solution further if I get λ=0 or λ=1

## The Attempt at a Solution

I think the algorithm for the solution should be the following:

Check if the sequence converges for α=1 by substituting α=1 in the equation, and solving for λ.

- If λ turns out to be 0<λ<1, then the order of convergence is 1.

- If λ turns out to be λ=0, λ=1, or 1<λ<∞, then don't play further with the solution.

- If λ turns out to be λ=∞, it means that the sequence diverges for α=1. Hence we must next substitute α=2 in the equation, and solve again for λ.

- If λ turns out to be 0<λ<1, then the order of convergence is 2.

- If λ turns out to be λ=0, λ=1, or 1<λ<∞, then don't play further with the solution.

- If λ turns out to be λ=∞, it means that the sequence diverges for α=2. Hence we must next substitute α=3 in the equation, and then solve again for λ. And so on.

I realize that there is a bug in the above algorithm. I only need someone to make me identify the bug.

Following is the solution according to the procedure stated above:

Substituting for α=1 in the equation:

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{|10^{-2^{n+1}}-0|}{|10^{-2^{n}}-0|^{1}}=λ[/itex]

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{10^{-2^{n+1}}}{10^{-2^{n}}}=λ[/itex]

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{10^{2^{n}}}{10^{2^{n+1}}}=λ[/itex]

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{10^{2^{n}}}{10^{2^{n}.2}}=λ[/itex]

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{10^{2^{n}}}{(10^{2^{n}})^2}=λ[/itex]

[itex]⇒\stackrel{limit}{_{n→∞}}\frac{1}{10^{2^{n}}}=λ[/itex]

[itex]⇒\frac{1}{∞}=λ[/itex]

[itex]⇒0=λ[/itex]

[itex]⇒λ=0[/itex]

... but according to the textbook (Burden Faries), the sequence has to quadratically converge to p=0. In other words, solving for λ with α=2 must give 0<λ<1, but it is clear that this is not going to happen.I just need to know what is wrong with my solution/algorithm.

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