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 4 of 4: Sum the degree-by-degree estimates
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∏i=1n(1+xi)=∑k=0nek≤1+∑k=1nSkk!\prod_{i=1}^n(1+x_i)=\sum_{k=0}^ne_k\le 1+\sum_{k=1}^n\dfrac{S^k}{k!}
Detailed analysis

Use e0=1e_0=1 and the upper bound from the previous step for k=1,…,nk=1,\ldots,n. The resulting inequality is exactly the required statement.