Partial Fraction Decomposition of unknown power

Question

I have an equation of the form: $$\frac{1}{(1+as)(1+0.5bs)^m}$$

where $m$ is unknown, and its range is $(1,2,3,...)$ How I can do the partial fraction?

I am now reading a paper that derived an equation like above. I attached the section of the paper that he did this, but he doesn't mention the steps. I want to know how this has been done.

Thank you very much.

Answer 1

For any $m$, the form of the partial fraction for your expression is $$ \frac{1}{(1+as)(1+0.5bs)^m} =\frac{c_0}{1 + as} + \sum_{i=1}^m \frac{c_i}{(1 + 0.5bs)^i}. $$

You determine the coefficients $c_0, c_1, \dots, c_m$ in the usual way: clear denominators to get a polynomial identity, then evaluate the polynomial identity or an appropriate derivative of the polynomial identity at a well-chosen value of $s$ to make all but two terms vanish.