series exercise

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EXERCISE: SERIES

1.

2.

a)

Derive the first three nonzero terms of Taylor’s Series of ln  3  x  centered at 2.

b)

 2 x  Expand ln  3  x  until the third term.

Given that P( x ) 

5 3  .Expand P( x ) as a series in ascending power of x , 2( x  1) 2( x  3)

up to the term in x 3 . 1 3 x)

3.

Derive the Maclaurin series for (1 

4.

a)

Give the Maclaurin series of the function e3x and cos(2x) .

b)

Find the first non-zero terms of Maclaurin series of e3 x cos( 2x ) .

5.

until the fourth term.

Show that by using Maclaurin series

1  xk

 1  kx 

kk  1 2 kk  1k  2 3 kk  1k  2k  3 4 x  x  x  ... 2! 3! 4! 7

6.

7.

 1  Ratio of the seventh coefficient to the fifth coefficient in  x 2   expansion is 1:20 px   when the series is written in ascending power of x. Determine the possible values of p.

 Without using calculator, find the exact value of

  4

5 1 

320

5 1

4

.

1

8.

a)

 1 x 3 3 Expand   until the term of x .  1  2x 

b)

Approximate the value of

3

11 by substituting x 

1 in question (a). 10 1

9.

Using binomial expansion, estimate the percentage of change in the gradient of y  x 3 if x is decreasing 3.5%.

10.

Each side length of a cube is decreased by 1.2%. Approximate the percentage of change in the volume and surface area.


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