If $a_1, \dots, a_n$ are all positive, show that $\prod_{k=1}^{n}(1+a_k) \leq \sum_{k=1}^{n+1}\frac{S^{k-1}}{(k-1)!}$, where $S = \sum_{k=1}^{n}a_k$
Prove that $\prod_{i=1}^n(1+x_i)\leq \sum_{i=0}^n\frac{S^i}{i!}$, where $x_i\in\mathbb{R^+}$.