Consider a Sequence of Rectangles $R _ { 1 }$ $R _ { 2 }$

Essay

Consider a sequence of rectangles, $R _ { 1 }$ , $R _ { 2 }$ , $R _ { 3 }$ , . . . illustrated in the figure below:
(a) The height $L _ { n }$ of $R _ { n }$ is given by $L _ { n } = f ( n )$ where $f ( x ) = \frac { 1 } { x }$ . Write down the first five terms of $\left\{ L _ { n } \right\}$ and determine the limit of $\left\{ L _ { n } \right\}$ .
(b) Let $b _ { n } = \sum _ { k = 1 } ^ { n } L _ { k }$ . Compare $b _ { n }$ to $\int _ { n } ^ { n + 1 } ( 1 / x ) d x$ .
(c) Determine whether $\left\{ b _ { n } \right\}$ converges or diverges. Justify your answer.