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

question 213

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.

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Definitions:

Hypertonic

A solution with a higher concentration of solutes compared to another solution, leading to osmosis.

Glucose Solution

A liquid preparation containing glucose, commonly used in medical settings for energy supply and fluid replacement.

Hydrostatic Pressure

The pressure exerted by a fluid at equilibrium due to the force of gravity, affecting every part of the fluid’s volume.

Passive Transport

The movement of substances across cell membranes without the need for energy input, utilizing concentration gradients.