2. FP2 (MEI) Maclaurin Series
Let Maths take you Further…

3. 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.

4. The general binomial expansion

5. Polynomial approximations

6. Consider f(x) = ex

7. When x=0
When x=0

11. For what values of x is this polynomial approximation valid for?

12. Autograph

13. In general terms….

14. Can you think of a ‘standard’ function of x such that the function and its derivatives evaluated at zero are undefined?

15. Can you explain why it is possible by referring to the graphs of lnx and ln(1+x)?

18. All Maclaurin expansions are centred on x=0
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)

21. Alternatively!!!!!

24. There are three methods to consider; we’ll work through the second method

27. 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.

28. 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.