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Putnam 1961 B5

Source: Putnam 1961

June 5, 2022
Putnaminequalitiesfactorial

Problem Statement

Let kk be a positive integer, and nn a positive integer greater than 22. Define f1(n)=n,    f2(n)=nf1(n),  …  ,fj+1(n)=nfj(n).f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}. Prove either part of the inequality fk(n)<n!!⋯!<fk+1(n),f_{k}(n) < n!! \cdots ! < f_{k+1}(n), where the middle term has kk factorial symbols.
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