2022 DHS Promo Q2
(a)
Given that $\text{f}$ is a continuous and increasing function, explain, with an aid of a sketch, why the value of
$\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left[ \text{f}\left( 1 \right)+\text{f}\left( 1+\frac{2}{n} \right)+\text{f}\left( 1+\frac{4}{n} \right)+…+\text{f}\left( 1+\frac{2\left( n-1 \right)}{n} \right) \right]$
is $\int_{1}^{3}{\text{f}\left( x \right)}\,\text{d}x$.
[2]
(b)
Hence, evaluate $\underset{n\to \infty }{\mathop{\lim }}\,\frac{2}{n}\left( {{\text{e}}^{2}}+{{\text{e}}^{2+\frac{4}{n}}}+{{\text{e}}^{2+\frac{8}{n}}}+…+{{\text{e}}^{2+\frac{4\left( n-1 \right)}{n}}} \right)$, leaving your answer in exact form.
[2]
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