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!
Worked Example
Maclaurin Series of Composites & Products
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
Worked Example
Differentiating & Integrating Maclaurin Series
How can I use differentiation to findMaclaurin 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