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limit of f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)) as n\to \infty

Source: Tuymaada 1996 p6

April 27, 2019
algebraSequenceLimit of sequencelimittrigonometric

Problem Statement

Given the sequence f1(a)=sin(0,5πa)f_1(a)=sin(0,5\pi a) f2(a)=sin(0,5π(sin(0,5πa)))f_2(a)=sin(0,5\pi (sin(0,5\pi a))) ...... fn(a)=sin(0,5π(sin(...(sin(0,5πa))...)))f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...))) , where aa is any real number. What limit aspire the members of this sequence as n→∞n \to \infty?
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