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1. The problem statement, all variables and given/known data

Sequence: x_{n}= (2x^{3}_{n-1}+ a)/3x^{2}_{n-1}. It is given that it converges to the cube root of a.

Verify that the order of convergence of this sequence is quadratic; i.e., verify that

lim absolute value(E_{n})/E^{2}_{n-1}

n->infinity

exists and is positive.

E represents the error in the nth term of the sequence; it is given that E_{n}= a^{1/3}- x_{n}

2. Relevant equations

Hint: consider (u+2v)(u-v)^{2}.

3. The attempt at a solution

1. The problem statement, all variables and given/known data

I plugged the expression for x_{n}into E_{n}, and used E_{n-1}= a^{1/3}- x_{n-1}. Then I substituted u = a^{1/3}and v = x_{n-1}and factored. My final limit:

lim abs((u+2v)(u-v)^{2})

n->inf. 3v^{2}(u-v)^{2}

My problem is that I have no idea how to take this limit using the variable n. Even if I had left this expression in terms of a^{1/2}and x_{n-1}I wouldn't know what to do, because I don't know how to work in how x changes as it's <i>subscript</i> goes to infinity.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Order of Convergence (mostly a limits question)

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