<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that $\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of</span></div><div class="list-item-paragraph" style="text-align: center">$\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\left[ \mathrm{f}\left( \frac{1}{n} \right)+\mathrm{f}\left( \frac{2}{n} \right)+...+\mathrm{f}\left( \frac{n}{n} \right) \right]$</div><div class="list-item-paragraph">is $\displaystyle\int\nolimits_{0}^{1}{\mathrm{f}(x)\mathrm{d}x}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Hence evaluate $\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1}{n}\left( \frac{\sqrt[3]{1}+\sqrt[3]{2}+...+\sqrt[3]{n}}{\sqrt[3]{n}} \right)$.</span></div><p style="text-align: right;">[3]</p></li></ol>

Given that $\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of

$\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\left[ \mathrm{f}\left( \frac{1}{n} \right)+\mathrm{f}\left( \frac{2}{n} \right)+...+\mathrm{f}\left( \frac{n}{n} \right) \right]$

is $\displaystyle\displaystyle \displaystyle\int\nolimits_{0}^{1}{\mathrm{f}(x)\mathrm{d}x}$.

Hence evaluate $\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1}{n}\left( \frac{\sqrt[3]{1}+\sqrt[3]{2}+...+\sqrt[3]{n}}{\sqrt[3]{n}} \right)$.

N2015 P1 Q3

N2015 P1 Q3

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