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Taylor Series using Geometric Series.

**Let **$\displaystyle f(x) = \frac{4-4x}{4x^{2} -8x -5};$ **given the partial decomposition,**

$\displaystyle \frac{4-4x}{4x^{2} -8x -5} = \frac{1}{5-2x} - \frac{1}{1+2x},$

**find the Taylor series of f(x) about 1. Express your answer in sigma notation and simplify as much as possible. Dtermine the open interval of convergence.**

See figure attached for my attempt.

Did I a mistake converting each piece into a taylor series by use of geometric series?

Also I can't think of how to simplify this anymore, but this may be due to a mistake in the first portion of my work.

Does anyone see any problems in my work?

Thanks again!