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The Symbolic Search for Iterations

In this section we show how a computer algebra package like MAPLE can be used to generate p-th order iterations that converge to

from the functions

. For a fixed initial value

and a fixed p we define the sequence

by

Of course, as it stands, although it is clear that

converges to

to p-th order, it is not very practical. The idea is to write the sequence recursively. Therefore there are two problems to solve:

Find initial values .
Get in terms of .

Table 1 contains some functions we defined in MAPLE and that we used to find and prove initial values and iterations.

Table 1: MAPLE functions

Our function etaq utilizes the expansion due to Euler:

We are then able to effectively compute q-series expansions of

,

,

. We have written MAPLE procedures to compute all the necessary functions.

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Contents Next: Initial values Up: Approximations to via the Previous: The functions