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Omni-MATH / Find, with proof, the smallest real number C with the following property:

Problem

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problem

Find, with proof, the smallest real number CC with the following property: For every infinite sequence {xi}\{x_i\} of positive real numbers such that x1+x2+⋯+xn≤xn+1x_1 + x_2 +\cdots + x_n \leq x_{n+1} for n=1,2,3,⋯n = 1, 2, 3, \cdots, we have \[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]
Plain-text mathematical notation (without MathML)
Find, with proof, the smallest real number C with the following property:
For every infinite sequence {x_(i)} of positive real numbers such that x₁+x₂+⋯+x_(n)≤x_(n+1) for n=1,2,3,⋯, we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}  \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]
Original LaTeX notation
Find, with proof, the smallest real number $C$ with the following property:
For every infinite sequence $\{x_i\}$ of positive real numbers such that $x_1 + x_2 +\cdots + x_n \leq x_{n+1}$ for $n = 1, 2, 3, \cdots$, we have
\[\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n}  \leq C \sqrt{x_1+x_2+\cdots+x_n} \qquad \forall n \in \mathbb N.\]

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Source and history

task_id
9c19a1ce-47ba-5918-a887-597037255cfa
task_key
test--9c19a1ce-47ba-5918-a887-597037255cfa
task_revision_id
1
config / split
/ test

Official source

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