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Riemann sum to evaluate limits question

Not sure how to get started on this question. Do I have to form the summation? If yes, how do I go about doing it? Any help will be really appreciated!


marked as duplicate by Martin Sleziak, Shailesh, Watson, Davide Giraudo, Ethan Bolker Nov 6 '16 at 15:48

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  • $\begingroup$ Please, post only one question in one post. Posting several questions in the same post is discouraged and such questions may be put on hold, see meta. $\endgroup$ – Martin Sleziak Nov 6 '16 at 8:44

Hint: For a) we have \begin{align} \lim_{n\rightarrow \infty}\sum^n_{k=1} \frac{k}{n^2+k^2}= \lim_{n\rightarrow \infty}\sum^n_{k=1} \frac{k/n}{n(1+k^2/n^2)} = \int^1_0 \frac{x}{1+x^2}\ dx. \end{align} Likewise, we can expression b) as an integral \begin{align} \int^1_0 \sin \pi x\ dx. \end{align}


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