the Further Mathematics network
www.fmnetwork.org.uk
the Further Mathematics network
www.fmnetwork.org.uk
FP2 (MEI)
Maclaurin Series
Let Maths take you Further…
Finding and using Maclaurin series
Before you start…
You need to have covered the work on inverse trigonometrical functions.
When you have finished…
You should:
Be able to find the Maclaurin series of a function, including the general term in
simple cases.
Appreciate that the series may converge only for a restricted set of values of x.
Identify and be able to use the Maclaurin series of standard functions.
The general binomial expansion
Polynomial approximations
Consider f(x) =
x
e
f ( x)  a0  a1 x
f ( x)  a0  a1 x  a 2 x
2
f ( x)  e
f ( x) 
f ( x) 
f '' ( x ) 
f '' ( x ) 
'
When x=0
'
When x=0
x
f ( x)  a0  a1 x  a 2 x 2  a3 x 3
f ( x)  a 0  a1 x  a 2 x 2  a3 x 3  a 4 x 4
f ( x)  a 0  a1 x  a 2 x 2  a3 x 3  a 4 x 4  .......  a r x r  ...
For what values of x is this polynomial approximation valid for?
Autograph
In general terms….
Can you think of a ‘standard’ function of x such that the function and its derivatives
evaluated at zero are undefined?
Can you explain why it is
possible by referring to the
graphs of lnx and ln(1+x)?
All Maclaurin expansions are centred on x=0.
But it is possible to form expansions centred
elsewhere. These latter two formulae are
alternative versions of the ‘Taylor
approximations’ centred on x=a. Notice
Maclaurin is a special case of a Taylor
approximation (using a=0)
Alternatively!!!!!
There are three methods to
consider; we’ll work through the
second method
Finding and using Maclaurin series
When you have finished…
You should:
•Be able to find the Maclaurin series of a function, including
the general term in simple cases.
•Appreciate that the series may converge only for a
restricted set of values of x.
•Identify and be able to use the Maclaurin series of
standard functions.
Independent study:

Using the MEI online resources complete the
study plan for Power series 1

Do the online multiple choice test for this and
submit your answers online.