Sum with Multi-Index Notation

Question

I have a doubt about sums with multi-index notation, maybe its pretty elementary but I need to clarify it: If $\alpha=(\alpha_1, \ldots, \alpha_n)\in\mathbb N^n\cup\{0\}$ can I write $$\sum_{j_1=0}^{\alpha_1}\cdots \sum_{j_n=0}^{\alpha_n}x^{j_1}\cdots x^{j_n}=\sum_{\beta\leq \alpha}x^\beta?$$

Answer 1

Yes, you can. (at least, provided that the notation $\beta\leqslant \alpha$ means that $\beta_i\leqslant \alpha_i$ for each $i\in [n]$ and $x^{\beta}=\prod_{j=1}^nx_j^{\beta_j}$).