Simplifying and Manipulating Summation

Question

Let $\tau=(\tau_1,\tau_2,\ldots)$ be a sequence of nonnegative integers which are finitely supported, i.e. only finitely many terms are nonzero. Let $$\rho=(\rho_j)_{j\geq 1}\in\mathbb{R}^{\mathbb{N}}.$$ We denote $|\tau|=\|\tau\|_{\ell^1}$, $\tau!=\prod_{j\geq 1}\tau_j!$ and $\rho^\tau = \prod_{j\geq 1}\rho_j^{\tau_j}$. How can we show the following: $$ \sum_{|\tau|=\ell}\frac{\rho^{\tau}}{\tau!} = \frac{1}{\ell!}\left(\sum_{j\geq 1}\rho_j\right)^{\ell} $$ My guess that it should be $"\leq"$, instead of equality but I tried examples such as $\ell=2$ and equality holds but I still cannot get the gist of this simplification. Thank you in advanced for the help.

Answer 1

You should assume $\sum|\rho_j|<\infty$ (for the two members of your equality to be defined).

The equality when only finitely many $\rho_j$'s are nonzero is merely the multinomial theorem.

The equality in the general case follows by passing to the limit.

q.e.d.