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Series of sum of cubes diverges

Source: Kvant Magazine No. 1 2020, 50th Anniversary, M365

March 10, 2023
real analysisConvergencedivergence

Problem Statement

[*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one? [*]The same question as before, for numbers not exceeding one. [*]Can it happen that the series a1+a2+⋯a_1+a_2+\cdots converges, but the series a13+a23+⋯a_1^3+a_2^3+\cdots diverges?
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