The diagram below shows the graph of $y={{2}^{x}}+1$ for $0\le x\le 1$. Rectangles, each of

Question

The diagram below shows the graph of $y={{2}^{x}}+1$ for $0\le x\le 1$. Rectangles, each of width $\frac{1}{n}$, are also drawn on the graph as shown. Show that the total area of all $n$ rectangles, ${{S}_{n}}$, is given by ${{S}_{n}}=\frac{{{2}^{\frac{1}{n}}}}{n\left( {{2}^{\frac{1}{n}}}-1 \right)}+1$[3] Find the exact value of ${\lim\limits_{n\to\infty} S_n }$.[2]

Accepted Answer

${\lim\limits_{n	o\infty} S_n = \frac{1}{\ln 2} + 1}$

2019 ACJC P1 Q6

2019 ACJC P1 Q6

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