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Nested Radical Sum

Source: Germany 2013/5

May 18, 2022
Upper Bound on SequencealgebraSequences

Problem Statement

Let α\alpha be a real number with α>1\alpha>1. Let the sequence (an)(a_n) be defined as an=1+2+3+…+n+n+1ααααa_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}} for all positive integers nn. Show that there exists a positive real constant CC such that an<Ca_n<C for all positive integers nn.
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