Question

Let ω0<π/2. The value of ∑[sin(ω0n)/πn]^4 equals

Answer 1

$$\displaystyle\sum_{i=1}^ \infin{(sin(nx))^4 \over n^4}\approx1.015(sinx)^{69/20}, 0<x<\pi/2$$

$$\displaystyle\sum_{i=1}^ \infin{(sin(n\omega_0))^4 \over n^4}\approx1.015(sin(\omega_0))^{69/20}, 0<\omega_0<\pi/2$$

Then

$$\displaystyle\sum_{i=1}^ \infin{(sin(\omega_0n))^4 \over (\pi n)^4}\approx{1.015 \over \pi^4}(sin(\omega_0))^{69/20}, 0<\omega_0<\pi/2$$