MathLabs

Problem 1

Let x1,x2,…,xnx_1,x_2,\ldots,x_n be positive real numbers and let S=x1+x2+⋯+xnS=x_1+x_2+\cdots+x_n. Prove that (1+x1)(1+x2)⋯(1+xn)≤1+S+S22!+S33!+⋯+Snn!(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 3 of 4: Obtain the upper bound for every symmetric sum
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ek≤Skk!(k=1,2,…,n)e_k\le\dfrac{S^k}{k!}\qquad(k=1,2,\ldots,n)
Detailed analysis

Rearranging Sk≥k!ekS^k\ge k!e_k gives ek≤Sk/k!e_k\le S^k/k!. This is exactly the estimate needed for each nonconstant term of the product expansion.