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Updated answer On v10.0.2, this achieves the same result as v8.0.4: ReleaseHold@Limit[ Hold[ Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, n}] ], n -> Infinity] (* 2/Pi *) Numerically, one can try this to investigate the behaviour of the function as n increases. func = Compile[{{n, _Integer}}, 2/Pi - Sum[Sin[Pi*k/n]/(n + 1/k), {k, 1, ...

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