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1 -1/k <= n (\sqrt[n]{k}-1 ) <= k - 1

Source: 1974 Swedish Mathematical Competition p2

March 26, 2021

inequalitiesalgebra

Problem Statement

Show that

1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1

for all positive integers

n

and positive reals

k