Is the function $f(x) = \sum_{n\geq 1 } \cos(n^2 x ) / n^{2.6}$ absolutely continuous?

Question

It is of course continuous.

Here the exponent $2.6 $ is delibrately chosen to be bigger than $2.5$.

If we differentiate the series formally term-by-term, we will get

$$ - \sum_n \frac{\sin n^2 x }{n^{0.6}} . $$

This series is at least in the $L^2 $ space. That is why a number bigger than $2.5$ is chosen. But I am not sure whether it is convergent. If in the numerator, it were $n$ in stead of $n^2$, it is convergent by the Dirichlet test.