By using the concept of $$\frac{1}{pq}=\frac{1}{q(p+q)}+\frac{1}{p(p+q)}$$ we can write $$\frac{1}{p^nq^m}=\sum_{i+j=m+n, i,j>o} \left[ \binom{j-1}{n-1}\frac{1}{q^i(p+q)^j}+ \binom{j-1}{m-1}\frac{1}{p^i(p+q)^j} \right]$$.

And also, we can write, $$\frac{1}{pqr}=\frac{1}{q(p+q)(p+q+r)}+\frac{1}{p(p+q)(p+q+r)}+\frac{1}{r(p+r)(p+q+r)}+\frac{1}{p(p+r)(p+q+r)}+\frac{1}{q(r+q)(p+q+r)}+\frac{1}{r(r+q)(p+q+r)}.$$

Then by using the above relations we can write the expression for $\frac{1}{p^nq^mr^l}$.

How can we extend this for the general case

$$\frac{1}{m_{1}^{k_1} m_{2}^{k_2} ... m_{r}^{k_r}} $$ in terms of $$\frac{1}{m_{i_1}^{j_1}(m_{i_1}+m_{i_2})^{j_2}...(m_{i_1}+...+m_{i_r})^{j_r}} $$


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