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)=ex
Examples
Find the first five terms of a power series for e3x+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