TITLE: Product of $n$ power series
QUESTION [5 upvotes]: I have $n$ power series. How can I find the power series of the product of these $n$ series? If there are two series $(a_m)$ and $(b_m)$ then the product series $(c_m)$ is given by the Cauchy product,
$$c_m = \sum_{k=0}^m a_k b_{m-k}$$
How does this generalize to more series?

REPLY [7 votes]: $$\left(\sum_{n=0}^\infty a_{1,n} x^n\right)\left(\sum_{n=0}^\infty a_{2,n} x^n\right)\dots\left(\sum_{n=0}^\infty a_{l,n} x^n\right) =  \sum_{n=0}^\infty  \sum_{k_1 + k_2 + \dots + k_l = n } a_{1,k_1}a_{2,k_2}\dots a_{l,k_l} x^n $$

REPLY [4 votes]: Presumably you are looking for an expression for the coefficients of $x^i$ in the product 
$$\left(\sum_j a_j x^j \right) \left(\sum_k b_k x^k \right) \left(\sum_l c_l x^l \right) \ldots \left(\sum_m n_m x^m \right)$$
Generalising the Cauchy product, this is   
$$\sum_i \left(\sum_{j=0}^{i} \sum_{k=0}^{i-j} \sum_{l=0}^{i-j-k} \cdots a_j b_k c_l \ldots n_{i-j-k-l-\cdots}  \right) x^i$$