Consider a sequence of rectangles,

R _ { 1 }

R _ { 2 }

R _ { 3 }

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

L _ { n }

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 }

\int _ { n } ^ { n + 1 } ( 1 / x ) d x

.
(c) Determine whether

\left\{ b _ { n } \right\}

converges or diverges. Justify your answer.

Find the Coefficient Of x3x ^ { 3 }x3 In the Binomial Series For

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Find the Coefficient Of $x ^ { 3 }$ In the Binomial Series For

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