To be honest I don't know how to start working with this series. I can't even tell if the limit of the general term is $0$. Any help would be appreciated.
$$\sum_{n=2}^{\infty}\left(\cos\left(\sqrt[3]{n^3+\sqrt{n}+7}\right)-\cos\left(\sqrt[3]{n^3-2\sqrt{n}+3}\right)\right).$$
For any $n$ big enough we have: $$\sqrt[3]{n^3+\sqrt{n}+7}-n \leq \frac{\sqrt{n}+7}{3 n^2}\leq \frac{1}{2n^{3/2}}$$ and: $$\sqrt[3]{n^3+\sqrt{n}+7} - \sqrt[3]{n^3-2\sqrt{n}+3} \leq \frac{3}{2n^{3/2}}.\tag{1}$$ Since the cosine function is Lipschitz continuous, the previous inequality implies: $$\left|\cos\sqrt[3]{n^3+\sqrt{n}+7} - \cos\sqrt[3]{n^3-2\sqrt{n}+3}\right|\leq\frac{3}{2n^{3/2}},\tag{2}$$ so the given series is absolutely convergent.