<s>[INST] Convergence of $\sum ( \cos \sqrt[3]{n^3 + \sqrt n + 7} - \cos \sqrt[3]{n^3 - 2\sqrt n + 3})$ I have some problem with this example: $$\displaystyle \sum_{n=2}^{\infty}\Bigg(\cos\Big(\sqrt[3]{n^3+\sqrt{n}+7}\Big) -\cos\Big(\sqrt[3]{n^3-2\sqrt{n}+3}\Big)\Bigg)$$
the only idea that crossed my mind is to use that $\cos x-\cos y=-2\sin\big({\frac{x+y}{2}}\big)\sin\big({\frac{x-y}{2}}\big)$ but later I don't know what to do with sines how to compare them or what else I can do with them ? 
 [/INST] Methodology


*

*Write $\cos x - \cos y = -2 \sin \frac{x+y}{2}\sin \frac{x-y}{2}$.

*Disregard the $-2$ and the $\sin \frac{x+y}{2}$, as $\lvert \sin \alpha \rvert \leq 1$

*Instead, focus on the $\sin \frac{x-y}{2}$ term. Intuitively, as $x \approx y$ when $n$ is large, we should expect this to get small, and therefore for this sine term to get small.

*Expand $x-y$ in power series to see that $x-y = \frac{1}{n^{3/2}} + \frac{4}{3n^2} + \dots$

*As $\sin x \approx x$ when $x$ is small, conclude that $\sin \frac{x-y}{2}$ behaves like $\frac{1}{2n^{3/2}}$

*Conclude that your series converges.

</s>