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sum 2^{k-1}/f_k <C for fermat numbers

Source: 2011 Saudi Arabia Pre-TST February 1.4

January 1, 2022
number theoryinequalitiesFermat numberalgebra

Problem Statement

Let fn=22n+1f_n = 2^{2^n}+ 1, n=1,2,3,...n = 1,2,3,..., be the Fermat’s numbers. Find the least real number CC such that 1f1+2f2+22f3+...+2n−1fn<C\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C for all positive integers nn
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