Let
$$s_n = \sum_{k=0}^{n}e^{ik^2}$$
where $i$ is the imaginary number. Is the sequence $s_n$ bounded in the complex plane ?
I have considered expanding the function $e^{z}$ at $z = ik^2$ which is in fact
$$e^{ik^{2}} = \sum_{j=0}^{\infty}{\frac{i^{j}k^{2j}}{j!}}$$
so that
\begin{align} s_n &= \sum_{k=0}^{n} \sum_{j=0}^{\infty} {\frac{i^{j}k^{2j}}{j!}} \\ &= \sum_{j=0}^{\infty} {\frac{i^jp_{2j}(n)}{j!}} \end{align}
where $p_{2j}(x) = 0^{2j}+1^{2j}+... +x^{2j}$. How can I proceed from here?