5.11.1 Maclaurin Series

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 we 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
• So, for example, 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

• This formula is in your exam formula booklet

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

• 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 actually having to find all the necessary derivatives of the function first
• STEP 2: Put the values from Step 1 into the general Maclaurin series formula
• STEP 3: Simplify the coefficients as far as possible for each of the powers of

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!
• 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

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

• A composite function is a ‘function of a function’ or a ‘function within a function’
• For example sin(2x) is a composite function, with 2x as the ‘inside function’ which has been put into the simpler ‘outside function’ sin x
• Similarly  is a composite function, with  as the ‘inside function’ and  as the ‘outside function’
• To find the Maclaurin series for a composite function:
• STEP 1: Start with the Maclaurin series for the basic ‘outside function’
• Usually this will be one of the ‘standard functions’ whose Maclaurin series are given in the exam formula booklet
• STEP 2: Substitute the ‘inside function’ every place that x appears in the Maclaurin series for the ‘outside function’
• So for sin(2x), for example, you would substitute 2x everywhere that x appears in the Maclaurin series for sin x
• STEP 3: Expand the brackets and simplify the coefficients for the powers of x in the resultant Maclaurin series
• This method can theoretically be used for quite complicated ‘inside’ and ‘outside’ functions
• On your exam, however, the ‘inside function’ will usually not be more complicated than something like kx (for some constant k) or xⁿ (for some constant power n)

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