Maclaurin Series of Standard Functions (DP IB Analysis & Approaches (AA)): Revision Note

Maclaurin series of standard functions

What is a Maclaurin Series?

• A Maclaurin series is a way of representing a function as an infinite sum of increasing integer powers of  ( etc.)

  • If all of the infinite number of terms are included, then the Maclaurin series is exactly equal to the original function

• If you truncate (i.e., shorten) the Maclaurin series by stopping at some particular power of , then the Maclaurin series is only an approximation of the original function

  • A truncated Maclaurin series will always be exactly equal to the original function for

  • In general, the approximation from a truncated Maclaurin series becomes less accurate as the value of moves further away from zero

  • The accuracy of a truncated Maclaurin series approximation can be improved by including more terms from the complete infinite series

    • E.g. a series truncated at the  term will give a more accurate approximation than a series truncated at the term

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

• Use the general Maclaurin series formula

• STEP 1
  Find the values of etc. for the function

  • E.g. for

    • 

    • 

  • An exam question will specify how many terms of the series you need to calculate (for example, “up to and including the term in ”)

  • You may be able to use your GDC to find these values directly, without having to find all the derivatives of the function first

• STEP 2
  Put the values from Step 1 into the general Maclaurin series formula

  • E.g. for

    • 

• STEP 3
  Simplify the coefficients as far as possible for each of the powers of

  • E.g. for

    • 

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 between Maclaurin series expansions and binomial theorem series expansions?

• Yes there is!

• For a function like  the binomial theorem series expansion is exactly the same as the Maclaurin series expansion for the same function

  • So unless a question specifically tells you to use the general Maclaurin series formula, you can use the binomial theorem to find the Maclaurin series for functions of that type

  • Or if you’ve forgotten the binomial series expansion formula for  where is not a positive integer, you can find the binomial theorem expansion by using the general Maclaurin series formula to find the Maclaurin series expansion