Series expansion of infinite series raised to the $n$th power So I know there is a well-known straightforward way to expand something like
$$(a+b)^n$$
and that there are formulas which allow us to expand trinomials and multinomials in general. My question is,

Is there any known way to expand something like $$\left[\sum_{k=0}^{\infty} a_k\right]^n$$ or at least to determine the first few terms?


If you know multinomials, then the result is the same: the series will contain infinite term, but each term will be composed by at maximum $n$ different $a_k$, and you can determine the coefficient of 
$a_{i_1}^{r_1}\dots a_{i_s}^{r_s}$, with $r_1+\dots+r_s = n$ by the multinomial
$$
\frac{n!}{r_1!\dots r_s!}
$$
