I want to show that $f(x)=\sum_{n\ge 1} \frac{\sin(nx)}{\sqrt n}\sim^{0^+} \sqrt{\frac \pi{2x}}$ by using Fourier series. I can't find the reasoning.
But i can show $f(x)\sim^{0^+} \sqrt{\frac \pi{2x}}$ using another way.
Proving that $f(x)\sim^{0^+}\int_0^{+\infty} \frac{\sin(t x)}{\sqrt t}dt$
notice that $\displaystyle \int_0^{+\infty} \dfrac{\sin(tx)}{\sqrt{t}}dt = \dfrac{1}{\sqrt{x}} \int_0^{+\infty} \dfrac{\sin(t)}{\sqrt{t}}dt = \sqrt{\dfrac{\pi}{2x}}$