Maclaurin’s  series

A power series centred at c may be written

1

When c = 0 , the power series is

2

let  

3
  

Differentiate and set x  to zero:-

4

5

And again,using a dot to show multiplication

6

7


and again,

8

9

and again,

10

11

etc.

Substituting back into the original series

12

Giving Maclaurin’s  series

13

14

Which has valid values for x when

15

 

Example

Find the Maclaurin expansion of

f(x) = (1+x)n

19

16

 

17

18

Which is the Binomial series

 

Example

Find the Maclaurin expansion of

f(x)=(1-x)n

20

 

Example

 

Use the Maclaurin Series to find a series for

f(x) = sinx

21

22

23

 

Example

Use the Maclaurin Series to find a series for

f(x) = (1+x)1/4

 

24

25

26

Example

Find the Maclaurin expansion of

f(x)=ex

27

28

 

29

30

Standard series

31

32

33

34

36

37

38

40

Examples

Find the first five terms of a power series for e3x+6

50

 

 

Find the first four terms of a power series for cos3x

41

Express  the following as a power series in x:-

42

43

 

Example

Calculate sin0.6 correct to five decimal places

44

45

 

 

 

© Alexander Forrest