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intergal of sin(nx)^n

Source: ICMC 2021 Round 1 P3

March 1, 2022
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Problem Statement

Let sn=∫01sinn(nx) dx\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx.
(a) Prove that sn≤2ns_n \leq \dfrac 2n for all odd nn.
(b) Find all the limit points of the sequence s1,s2,s3,…s_1, s_2, s_3, \dots.
Proposed by Cristi Calin
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