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On the supremum and infimum of a sine sequence

Let be a sequence defined as

Find the supremum as well as the infimum of the sequence .

Solution

Background: This problem was on the shortlist of the 2014 Olimpiada Nationala de Matematica de Romania and was suggested by Leo Giugiuc.

We begin by the very well known manipulation.

Thus and we have to find the supremum and infimum of . Since the values are dense on the unit circle , the same shall hold for implying that and . Thus,

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