Study the series of $\int_0^{\frac{1}{n^a}}\sin{(\sqrt[3]{x})}\,dx$ with respect to $a>0$ I have to study the series $\sum a_n$ with $a_n=\int_0^{\frac{1}{n^a}}\sin{(\sqrt[3]{x})}\,dx$, with respect to $a>0$.
I have thought to use the asymptotic criterion for series.
In particular I can observe that:
$\sin{x}\sim x-\frac{x^3}{3!}$ and so: $\sin{(\sqrt[3]{x})}\sim \sqrt[3]{x}- \frac{x}{3!}$ for $x\to 0$.
Now this means that: $$\int_0^{\frac{1}{n^a}}\sqrt[3]{x}- \frac{x}{3!}\,dx=[\frac{x^{\frac{4}{3}}}{\frac{4}{3}}]\rvert_0^{\frac{1}{n^a}}-[\frac{x^{2}}{2\cdot 3!}]\rvert_0^{\frac{1}{n^a}}=\frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}-\frac{1}{12n^{2a}}$$ Now since $\frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}-\frac{1}{12n^{2a}}$ it self is asymptotic to $\frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}$ then I can say:
$$a_n\sim \frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}-\frac{1}{12n^{2a}}\sim \frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}$$
Now $\sum_{n=1}^{\infty} \frac{1}{\frac{4}{3}n^{\frac{4}{3}a}}$ is convergent for $\frac{4}{3}a>1\iff a>\frac{3}{4}$ and so from asymptote criterion also the original series converges for this value of $a$.
I need a check of my attempt and in case there is something wrong can you tell me where and how can I correct this?
(I can't use the Lebesgue theore since it is not something that I know)

Here is a more rigourous version of what you did. Using the Lagrange form of the remainder term in Taylor's formula, we have
$$
\left| {\sin w - w} \right| \le \frac{{w^3 }}{6}
$$
for all $w\geq 0$. Consequently,
\begin{align*}
\left| {a_n  - \frac{3}{{4n^{4a/3} }}} \right| &= \left| {\int_0^{1/n^a } {(\sin \sqrt[3]{x} - \sqrt[3]{x})dx} } \right| \\ & \le \int_0^{1/n^a } {\left| {\sin \sqrt[3]{x} - \sqrt[3]{x}} \right|dx}  \le \int_0^{1/n^a } {\frac{x}{6}dx}  = \frac{1}{{12n^{2a} }}.
\end{align*}
Thus,
$$
\left| {\frac{{a_n }}{{\frac{3}{{4n^{4a/3} }}}} - 1} \right| \le \frac{1}{{9n^{2a/3} }}
$$
showing that
$$
a_n  \sim \frac{3}{{4n^{4a/3} }}
$$
as $n\to +\infty$.
Addendum. By L'Hôpital's rule
$$
\mathop {\lim }\limits_{w \to 0} \frac{{\int_0^w {\sin \sqrt[3]{x}dx} }}{{\frac{3}{4}w^{4/3} }} = \mathop {\lim }\limits_{w \to 0} \frac{{\sin \sqrt[3]{w}}}{{\sqrt[3]{w}}} = 1
$$
In particular, with $w=1/n^a$,
$$
a_n  \sim \frac{3}{{4n^{4a/3} }}
$$
as $n\to +\infty$.
