I try to use the following result:
If $f(x)\in C^1[1,+\infty)$ and $\displaystyle\int_1^{+\infty} |f'(x)|{\rm d}x$ is convergent, then $\displaystyle\sum_{n=1}^{\infty} f(n)$ has the same convergence or divergence as $\displaystyle\int_1^{+\infty} f(x) {\rm d}x$.
Now, define $f(x):=\dfrac{\sin x^{3/2}}{x^{1/2}}$. Then \begin{align*} \int_1^{+\infty}|f'(x)|{\rm d}x&=\int_1^{+\infty}\left|\frac{3}{2}\cos x^{3/2}-\frac{\sin x^{3/2}}{2x^{3/2}}\right|{\rm d}x\\ &=\int_1^{+\infty}\left|\frac{\cos x}{x^{1/3}}-\frac{\sin x}{3x^{4/3}}\right|{\rm d}x\\ &\ge \int_1^{+\infty}\frac{|\cos x|}{x^{1/3}}-\frac{|\sin x|}{3x^{4/3}}{\rm d}x=+\infty, \end{align*}
which is not convergent.
