I would like to find the sum of the series
$$ \sum u_{n}$$
where $$ u_{n}=(-1)^n\int_0^1 \cos(nt^2)\mathrm dt$$
Using the change of variable $t\rightarrow \sqrt{n}t$:
$$ u_{n}=\frac{(-1)^n}{\sqrt{n}} \int_0^{\sqrt{n}} \cos(t^2)\mathrm dt\sim_{n\rightarrow \infty} \frac{(-1)^n}{2}\sqrt{\frac{\pi}{2n}}$$
So $\sum u_{n}$ is convergent.
What about
$$ \sum_{n=0}^{\infty} (-1)^n\int_0^1 \cos(nt^2)\mathrm dt$$
?