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Omni-MATH / Determine the maximum value of the sum

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Determine the maximum value of the sum S=∑n=1∞n2n(a1a2⋯an)1/nS = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n} over all sequences a1,a2,a3,⋯a_1, a_2, a_3, \cdots of nonnegative real numbers satisfying ∑k=1∞ak=1.\sum_{k=1}^\infty a_k = 1.
Plain-text mathematical notation (without MathML)
Determine the maximum value of the sum
S=∑_(n=1)^(∞)(n)/(2^(n))(a₁a₂⋯a_(n))^(1/n)
over all sequences a₁,a₂,a₃,⋯ of nonnegative real numbers satisfying
∑_(k=1)^(∞)a_(k)=1.
Original LaTeX notation
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\]

Discussion

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Source and history

task_id
082bc42c-4449-5fb6-a795-4a01989f1a68
task_key
test--082bc42c-4449-5fb6-a795-4a01989f1a68
task_revision_id
1
config / split
/ test

Official source

initial import