First of all, we observed the image of sinnx/x^2, and found that it was alternating with peaks and valleys, and the width of peaks and valleys remained unchanged. The area of the first peak is larger than that of the valley behind it. So we find that the absolute value of the integral formula is less than the maximum of the area of the first peak and the area of the first valley.

From the above observations, we give proof.

S(n)=|∫+∞ 1 (sinnx/x*x)|

Set 2 kpis.

s(n)& lt； =max{ 2kPi to 2k+ 1Pi under integer, sinnx/x 2-(area of the first peak)

Under integer, 2k+ 1Pi to 2k+2Pi, -sinnx/x 2-(the area of the first valley).

}

Under integer, 2kPi to 2k+ 1Pi, sinnx/x 2.

Under the integer symbol, 2k+ 1Pi to 2k+2Pi, -sinnx/x 2 make the same estimation.

Therefore s (n)->; 0, (n tends to infinity)

Note: The above observations can be proved by analytical means, but they are trivial and will not be typed.