<s>[INST] Find the given limit of $S$? 
Find the given limit  of $S$ ?
$$S=\lim _{n \rightarrow \infty}  \left (\frac{\sin \frac{\pi}{ n+1}} { 1} +\frac{\sin \frac{2\pi}{ n+1}} { 2} +  + ....+ \frac{\sin \frac{n\pi}{ n+1}} { n} \right )$$

My attempt :
i construct  modified the  given series
$\frac{\pi}{ n+1}\left (\frac{\sin \frac{\pi}{ n+1}} { \frac{1\pi}{ n+1}} +\frac{\sin \frac{2\pi}{ n+1}} { \frac{2\pi}{ n+1}} + ... + ....+ \frac{\sin \frac{n\pi}{ n+1}} { \frac{n\pi}{ n+1}} \right )$
After that im not able to proceed further
Any hints/solution will be appreciated
thanks u
 [/INST] The limit exists and is bounded above as
$$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k}\sin  \left( \frac{k\pi}{n+1}\right) \leqslant \lim_{n \to \infty} \sum_{k=1}^n  \frac{\pi}{n+1} = \lim_{n \to \infty} \frac{n}{n+1} \pi = \pi$$
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