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 calculator 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 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
Worked example
Use the Maclaurin series formula to find the Maclaurin series for up to and including the term in .