Problem 1

Let

x_1,x_2,\ldots,x_n

be positive real numbers and let

S=x_1+x_2+\cdots+x_n

. Prove that

(1+x_1)(1+x_2)\cdots(1+x_n)\le 1+S+\dfrac{S^2}{2!}+\dfrac{S^3}{3!}+\cdots+\dfrac{S^n}{n!}

Step 1 of 4: Introduce the elementary symmetric sums

Previous step Next step

e_k=\sum_{1\le i_1<\cdots<i_k\le n}x_{i_1}\cdots x_{i_k},\qquad\prod_{i=1}^n(1+x_i)=\sum_{k=0}^n e_k

Detailed analysis

Let $e_k$ be the sum of all products of $k$ distinct variables, with $e_0=1$ . Choosing either $1$ or $x_i$ from each factor gives the product expansion $\prod(1+x_i)=e_0+e_1+\cdots+e_n$ .