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smallest k such that 1 \le n ^{1/n} \le k for all positive integers n

Source: Brazilian Mathematical Olympiad 1983 P5

July 27, 2018
number theoryminimumpositive integersInequality

Problem Statement

Show that 1≤n1/n≤21 \le n^{1/n} \le 2 for all positive integers nn. Find the smallest kk such that 1≤n1/n≤k1 \le n ^{1/n} \le k for all positive integers nn.
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