Let \(x=(x_1,\ldots,x_n)\in \mathbb{R}^n\) and \(m \in \mathbb{N}\). Then

\begin{equation*} (x_1+\cdots +x_n)^m=\sum_{|\alpha|=m} \binom{m}{\alpha} x^{\alpha}. \end{equation*}

Note, we used multi-index notation and \(\binom{m}{\alpha} \) denotes the multinomial coefficient .

See also Link to heading

• binomial theorem
• For \(a,b\ge 0\) and \(k\in \mathbb{N}\) \[ (a+b)^k\le 2^k(a^k+b^k) \] (see (0x68bbccca) ).