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Ive got another question:

Express the limit $lim_{n\to\infty}\sum_{k=1}^n\frac{\pi}{n}sin(\frac{\pi k}{n})$ as a Riemann Integral

I have the knowledge of Riemann Integrals but cant apply it to this question. Any help will be appreciated.

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  • $\begingroup$ Can you give us the definition of a Riemann integral? $\endgroup$ – Doug M Nov 9 '16 at 2:49
  • $\begingroup$ Well a Riemann Integral can be defined as: $\int_a^bf(x)dx=lim_{n\to\infty}\sum_{k=1}^{n}f(x_i)\delta{x}$ $\endgroup$ – user384716 Nov 9 '16 at 2:59
  • $\begingroup$ and isn't what you have? $[x_i,x_{i+1}] = [\frac {\pi i}{n}, \frac {\pi (i+1)}{n}], \delta x = \frac {\pi}{n}$ $\endgroup$ – Doug M Nov 9 '16 at 3:12
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Observe \begin{align} \sum^n_{k=1} \frac{\pi}{n} \sin\left(\frac{\pi k}{n} \right) \approx \int^1_0 \pi \sin \pi x\ dx. \end{align}

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