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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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Previous: The Quadratic and