Converges or Diverges

Consider $\large\mathfrak Y= \displaystyle\lim_{r\to\infty}\displaystyle \sum_{n=1}^r \left[n \sin \left(\frac1n \right) - (n+2) \sin \left(\frac1{n+2} \right) \right]$

The series is obviously a telescopic one which decomposes to: $\sin(1)+2\sin\left(\dfrac 12\right)-\displaystyle\lim_{r\to\infty}\left((r+1)\sin\left(\dfrac1{r+1}\right)\right)-\displaystyle\lim_{r\to\infty}\left((r+2)\sin\left(\dfrac1{r+2}\right)\right)$

For limit calculation, use $\displaystyle\lim_{x\to 0}\dfrac{\sin x}x=1$ , so that sum converges to:

$\large\boxed{\color{#0C6AC7}{\mathfrak Y=\sin(1)+2\sin\left(\dfrac 12\right)-2}}$