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The Cubic Iteration

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  We summarise the results of [6,7] and how they correspond to the case p=3. Define

   

Then

    

We also mention the identity

 

This equation was known to Ramanujan [3, p. 199,]. See [2] for a proof using theta-functions.

We have

   

As usual we let

 

Then, from (6.9)--(6.12) we have

  

and

 

Now, from (6.7), (6.13), (6.14) we have

 

and, by (3.18), (6.16) we have

 

where

 

From (3.19), (3.20) we know that

  

and so

 

By letting,

   

we have the following

Theorem 6.1

(Borwein and Borwein [6, Iteration 3.2, p. 699,]) Define sequences , , and by

Then converges cubically to .

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