Section 11.4 – Representing Functions with Power Series

10.5

4. Find the power series representation for

centered about x = 0 and specify its radius of convergence.

1f x

3 2x

1/ 31 1/ 3f x

23 2x1

1/ 3x

3

Infinite geometric with first term 1/3 and r = -2x/3

Converges when |r| < 1

3radi

2

2

xs1

3u

8. Find the power series representation for

centered about x = 0 and specify its radius of convergence.

x

20

2f x dt

1 t

x

2 4 6

0

f x 2 2t 2t 2t ... dt

3 5 72x 2x 2x

f x 2x ...3 5 7

2 n

2 1n

1 1

n

2x 2n 1lim 1

2 1 2n 1 x

2

n

2n 1x lim 1

2n 1

2x 1

Radius of Convergence is 1

2n 12x

2n 1

26.Use an appropriate identity to find the Maclaurin series for f(x) = sin x cos x

1 1f x sinx cosx f x 2sinx cosx f x sin2x

2 2

3 5 7 9x x x xsinx x ...

3! 5! 7! 9!

3 5 7 92x 2x 2x 2x

sin 2x 2x ...3! 5! 7! 9!

2 3 4 5 6 7 8 91 2 x 2 x 2 x 2 x

sin 2x x ...2 3! 5! 7! 9!

2n 2n 1

n

n 0

2 x1

2n 1 !

30. Given the function f defined by f x 4 x

a. Find the first three nonzero terms in the Maclaurin series for the function f.

3 / 2

f x 4 x f 0 2

1 1f ' x f ' 0

42 4 x1 1

f '' x f '' 0324 x 4

22p x x

1 1/x2

!

32

4 2

22

1 1p x 2 x x

4 64

30. Given the function f defined by f x 4 x

b. Find the first three terms in the Maclaurin series for the

function g defined by 3g x 4 x

22

1 1p x 2 x x

4 64

32

3 3 2x x

1 1p 2

4 6x

4

3 3 62

1 1p x 2 x x

4 64

30. Given the function f defined by f x 4 x

c. Find the first four terms in the Maclaurin series for the

function h defined by 34 x and 2h' 0x h

3 61 12 xh' x x

4 64

3 61 12 x x dx

4 6h' x

4

4 71 12x x x

16 4h

8x C

4

4 71 12 2x x x

16 44x

8h

2nn

n 0

n 10 n

Let f be the function defined by the power series f x = a x

awhere a 1and a for n 1

na. Write the first five terms of the series and the general term.

b. Determine the radius of convergence for

the series in part (a).

Show the work that leads to your answer.

c. Show that f' x 2xf x

2n

2 4 6 81 1 1 xf x 1 1x x x x ...

2 6 24 n!

2n 2

2nn

x n!lim 1

n 1 ! x

2

n

xlim 1

n 1

0 1

Converges for all x

2nn

n 0

n 10 n

Let f be the function defined by the power series f x = a x

awhere a 1and a for n 1

nc. Show that f ' x 2xf x

2n

2 4 6 81 1 1 xf x 1 1x x x x ...

2 6 24 n!

2n 1

3 5 7 91 1 2x2xf x 2x 2x x x x ...

3 12 n!

2n 1

3 5 7 91 1 2nxf ' x 2x 2x x x x ...

3 12 n!

2n 1 2n 1

3 5 7 91 1 2x 2xf ' x 2x 2x x x x ...

3 12 n 1 ! n!