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More analysis... Find a bound for sum of sqrt[k]{x}

Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 3

October 21, 2023
real analysisinequalitiesalgebra

Problem Statement

Prove that there exists a constant C>0C > 0 such that, for any integers m,nm, n with n≥m>1n \geq m > 1 and any real number x>1x > 1, ∑k=mnxk≤C(m2⋅xm−1log⁡x+n)\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)
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