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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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Next: The Septic Iteration
Up: Approximations to via the
Previous: The Quadratic and