<p>In this question, \(a\) is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Explain, with the aid of a sketch, why\n\n\[\frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) &gt; \int_{0}^{a} \ln(1+x) \, \mathrm{d}x\] for all positive integers \(n\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">State a similar expression that is less than \(\int_{0}^{a} \ln(1+x) \, \mathrm{d}x\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Use your sketch in part <strong>(i)</strong> to explain why \[\lim_{n \to \infty} \left[ \frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) \right] = \int_{0}^{a} \ln(1+x) \, \mathrm{d}x\] and find the value of this limit exactly in terms of \(a\).<span class="inline-marks">[5]</span></div></li></ol>

In this question, \(a\) is a positive constant.

Explain, with the aid of a sketch, why\n\n\[\frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) > \int_{0}^{a} \ln(1+x) \, \mathrm{d}x\] for all positive integers \(n\).

State a similar expression that is less than \(\int_{0}^{a} \ln(1+x) \, \mathrm{d}x\).

Use your sketch in part **(i)** to explain why \[\lim_{n \to \infty} \left[ \frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) \right] = \int_{0}^{a} \ln(1+x) \, \mathrm{d}x\] and find the value of this limit exactly in terms of \(a\).

Applications of Intergration - Riemann Sums

Applications of Intergration - Riemann Sums