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!