Maclaurin Series (Edexcel A Level Further Maths: Core Pure)

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Roger

Last updated

2 September 2022

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 $x$ ( $x^{1}, x^{2}, x^{3},$ 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 $x$ , 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 $x = 0$
In general, the approximation from a truncated Maclaurin series becomes less accurate as the value of $x$ 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 $x^{7}$ term will give a more accurate approximation than a series truncated at the $x^{3}$ term

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

Use the general Maclaurin series formula

$f (x) = f (0) + x f' (0) + \frac{x^{2}}{2!} f'' (0) + . . .$

This formula is in your exam formula booklet

STEP 1: Find the values of $f (0), f' (0), f'' (0),$ 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 $x^{4}$ ”)
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 $x$

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

Yes there is!
For a function like ${(1 + x)}^{n}$ 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 ${(1 + x)}^{n}$ where $n$ 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

f (x) = \sqrt{1 + 2 x}

up to and including the term in

x^{4}

5-11-1-ib-aa-hl-maclaurin-series-standard-a-we-solution

Maclaurin Series of Standard Functions

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:

$e^{x} = 1 + x + \frac{x^{2}}{2!} + . . . \frac{x^{r}}{r!} + . . .$ valid for all $x$

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - . . . + {(- 1)}^{r + 1} \frac{x^{r}}{r} + . . .$ valid for $- 1 < x \leq 1$

$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - . . . + {(- 1)}^{r} \frac{x^{2 r + 1}}{(2 r + 1)!} + . . .$ valid for all $x$

$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - . . . + {(- 1)}^{r} \frac{x^{2 r}}{(2 r)!} + . . .$ valid for all $x$

$\arctan x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - . . . + {(- 1)}^{r} \frac{x^{2 r + 1}}{2 r + 1} + . . .$ valid for $- 1 \leq x \leq 1$

${(1 + x)}^{n} = 1 + n x + \frac{n (n - 1)}{1 \times 2} x^{2} + . . . + \frac{n (n - 1) . . . (n - r + 1)}{1 \times 2 \times . . . \times r} x^{r} + . . .$ valid for $| x | < 1$

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

Maclaurin Series of Compound Functions

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 $e^{x^{2}}$ is a composite function, with $x^{2}$ as the ‘inside function’ and $e^{x}$ 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

Worked example

Find the Maclaurin series for the function

f (x) = \ln (1 + 3 x)

, up to and including the term in

x^{4}

3-3-1-fm-maclaurin-series-of-comp-funct-a-we-solution

Find the Maclaurin series for the function

g (x) = e^{x} \sin x

, up to and including the term in

x^{4}

3-3-1-fm-maclaurin-series-of-comp-funct-b-we-solution

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