I have the following:
$$ y = \sum_{n=0}^\infty [x^n \sum_{k=0}^\infty (k+1)a_{k+1} P_{n-k}] \sum_{n=0}^\infty x^n[s_n - \sum_{k=0}^n a_{k+1}(k+1)R_{n-k}] $$
I can easily multiply
$$ \sum_{n=0}^\infty [a_n x^n] \sum_{n=0}^\infty b_nx^n = \sum_{n=0}^\infty x^n \sum_{k=0}^\infty (a_k b_{n-k}) $$ But the summation above i seem to be lacking the intuition to find the outcome of the summation, What is a trick to finding out the result of the summation?