approximation by series expansion

Hi everybody!

I try to approximate the term 1/(a-x)
ar 1/(a-cos(theta)) for a>1.
In order to approximate it I want to expand it in a power series.
Is there a rule of thumb which kind of expansion will give me the
best convergence in the region x=[0,1]?
It is easy to get good convergence for moderate a. However, for a close to 1
(e.g. 1.01 the convergence is quite bad)
Is there a reasonable way to make the expansion be exact at x=0 and x=1 at
the same time?

Thank You,

Andreas
.

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