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I want to use MMA to find the limit of $\lim _{n \rightarrow \infty}\left(\frac{\sin \frac{\pi}{n}}{n+1}+\frac{\sin \frac{2 \pi}{n}}{n+\frac{1}{2}}+\cdots+\frac{\sin \pi}{n+\frac{1}{n}}\right)$.

Limit[Sum[Sin[(i*Pi)/n]/(n + 1/i), {i, 1, n}], n -> Infinity]

However, the above code does not get the correct result (the answer is $\frac{2}{\pi}$).

What can I do to get the right results?

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1 Answer 1

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The recent command of Mathematica

AsymptoticSum[Sin[(i*Pi)/n]/(n + 1/i), {i, 1, n}, {n, Infinity, 1}]
(*2/\[Pi]*)

does the job.

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