In this question, $a$ is a positive constant. Explain, with the aid of a sketch, why \(\frac{a}{n}

Question

In this question, $a$ is a positive constant. Explain, with the aid of a sketch, why $\frac{a}{n} \sum_{k=1}^{n} \ln \left(1 + \frac{ak}{n}\right) > \int_{0}^{a} \ln(1+x) \, \mathrm{d}x$ for all positive integers $n$.[2] State a similar expression that is less than $\int_{0}^{a} \ln(1+x) \, \mathrm{d}x$.[1] 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 $a$.[5]

Accepted Answer

(i) The right Riemann sum is greater than the integral (ii) $\dfrac an\sum_{k=0}^{n-1}\ln\left(1+\dfrac{ak}{n}\right)<\displaystyle\int_0^a\ln(1+x)\,\mathrm dx$ (iii) $(a+1)\ln(a+1)-a$

Applications of Integration - Riemann Sums

Applications of Integration - Riemann Sums

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