IB DP Maths: AA HL复习笔记5.11.1 Maclaurin Series

What is a Maclaurin Series?

How do I find the Maclaurin series of a function ‘from first principles’?

Is there an easier way to find the Maclaurin series for standard functions?

• Yes there is!
• The following Maclaurin series expansions of standard functions are contained in your exam formula booklet:

• Unless a question specifically asks you to derive a Maclaurin series using the general Maclaurin series formula, you can use those standard formulae from the exam formula booklet in your working

Is there a connection Maclaurin series expansions and binomial theorem series expansions?

• Yes there is!

How can I find the Maclaurin series for a composite function?

How can I find the Maclaurin series for a product of two functions?

• To find the Maclaurin series for a product of two functions:
  • STEP 1: Start with the Maclaurin series of the individual functions
    • For each of these Maclaurin series you should only use terms up to an appropriately chosen power of x (see the worked example below to see how this is done!)
  • STEP 2: Put each of the series into brackets and multiply them together
    • Only keep terms in powers of x up to the power you are interested in
  • STEP 3: Collect terms and simplify coefficients for the powers of x in the resultant Maclaurin series

How can I use differentiation to find Maclaurin Series?

• If you differentiate the Maclaurin series for a function f(x) term by term, you get the Maclaurin series for the function’s derivative f’(x)
• You can use this to find new Maclaurin series from existing ones
  • For example, the derivative of sin x is cos x
  • So if you differentiate the Maclaurin series for sin x term by term you will get the Maclaurin series for cos x

How can I use integration to find Maclaurin series?

• If you integrate the Maclaurin series for a derivative f’(x), you get the Maclaurin series for the function f(x)
  • Be careful however, as you will have a constant of integration to deal with
  • The value of the constant of integration will have to be chosen so that the series produces the correct value for f(0)
• You can use this to find new Maclaurin series from existing ones
  • For example, the derivative of sin x is cos x
  • So if you integrate the Maclaurin series for cos x (and correctly deal with the constant of integration) you will get the Maclaurin series for sin x