Maclaurin’s  series

A power series centred at c may be written

When c = 0 , the power series is

let  

  

Differentiate and set x  to zero:-

And again,using a dot to show multiplication

and again,

and again,

etc.

Substituting back into the original series

Giving Maclaurin’s  series

Which has valid values for x when

 

Example

Find the Maclaurin expansion of

f(x) = (1+x)ⁿ

 

Which is the Binomial series

 

Example

Find the Maclaurin expansion of

f(x)=(1-x)ⁿ

 

Example

 

Use the Maclaurin Series to find a series for

f(x) = sinx

 

Example

Use the Maclaurin Series to find a series for

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

 

Example

Find the Maclaurin expansion of

f(x)=e^x

 

Standard series

Examples

Find the first five terms of a power series for e^3x+6

 

 

Find the first four terms of a power series for cos3x

Express  the following as a power series in x:-

 

Example

Calculate sin0.6 correct to five decimal places