Trigonometric sums and limits

Calculus Level 5

Given that \[S(n)=\displaystyle\sum_{k=1}^{n} \sin {kx},\]

find the value of

\[\displaystyle\lim_{x \rightarrow \pi/3} \lim_{p \rightarrow \infty}\dfrac{ S(1)+S(2)+S(3)+......+S(p)}{p}.\]

Note: This is not an original question - it was provided by our professor.

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