A little correction
means the union of the sets . is not defined on sets. Then, you are dividing by . The notation you are using is meaningless. The only place you define is in the sum. But then you use outside the sum, as well. Nowhere do you have anything that would allow partial fractions to be applicable.
Edit: from the series you tried to write, it looks like you are looking for:
If that is what you mean, then what do you think partial fractions are?