Let $\displaystyle k(x) = \sum_{n=1}^{\infty}\frac{\sin{(nx^2)}}{3^{n}}$. Prove $\displaystyle k$ is continuous on $\displaystyle \mathbb{R}$.

NOTE: for this proof, you're allowed to assume $\displaystyle \sin{(x)}$ is continuous on $\displaystyle \mathbb{R}$.