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# §11.9 Representations of Functions as Power Series

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1      國立交通大學應用數學系 莊重教授

1 ... , | | 1.

1 1

0

2

x x x

x x n

n

Example 1：

1 , 1 | | 1. 1

1 1

1 2 2

0 0

2 2

2

x x

x x x

x

n n

n n

n

Example 2：

2.

|

| 2 , 2

2 1 1 2

1 2 1 2

1

0 1 0

x x x

x x n n

n

n n

Example 3：

0 1

3

0 3 3

3

x x x

x x x

x x

n

n n n n

n

## 2. 有時須微分或積分不只一次才到直系。

(2)

2      國立交通大學應用數學系 莊重教授

Example 4：

Solution：

. 1

|

| ..., 1 1

1

... 1 3 1 2

ln

2 0

0 1 3

2

x x

x x x

n c x x

x x c x

n n

n n

dx

d



1 0

1

n x x n x x x

n n n

n

Example 5：

f(x)tan1x.

0 0

tan 1

) (

n n n

x tdt a x

x

f

a8 ?

Solution：

0

2 2

0

2 1

x x x

x x c

x

n

n n n

n n

dx

d

-1

c

0

1 2

1

n x x x

n

n n

Example 6：

## Find the power series representation of

1

### 

.

) 1

( 2

x x

f

(3)

3      國立交通大學應用數學系 莊重教授

Solution：

. 1

|

| 1 ,

1

. 1

|

| , 1

1

0 1

1 2

x x x

x nx x

n n dx d n

n



Example 7：

x7 dx

Solution：

0

1 7 0

7 7 0

7

x n c

x dx x

dx x x dx

n

n n n

n n n n

n

Example 8：

?, | | 1.

1

1

x

nx

n n

?, | | 1.

1

x nx

n n

?

12

n

n

n

1

?, | | 1.

2

x x

n n

n

n

?

2 2

2

n

n

n n

## (iii)

?

12

2

### 

n

n

n

(4)

4      國立交通大學應用數學系 莊重教授

Solution：

2 1

1 1

1

x x dx x d

dx x d dx nx d

n n n

n n

n

2

1 1

1 1 x

nx x x nx

n n n

n

2

1

n

n

n

2

1

n

xn

n n

3

2

2 1 2

2

1 2

2

2 2

x nx x

dx x d dx nx

x d x n n x

n n n

n n

n

3

2

2

2

n

n

n n

## (iii)

4 2 6.

2 2

2 2

2 2 1

2

1 1

2

1

2

### 

n

n n

n n

n n

n n

n

n n

n n

n n n

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