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

• STEP 1
  Find the values of $f\left(0\right),f^{\text{'}}\left(0\right),f^{\text{'}\text{'}}\left(0\right),$ etc. for the function

  • E.g. for $f\left(x\right)=\sin x$

    • $f^{\text{'}}\left(x\right)=\cos x,f^{\text{'}\text{'}}\left(x\right)=-\sin x,f^{\text{'}\text{'}\text{'}}\left(x\right)=-\cos x$

    • $f\left(0\right)=0f^{\text{'}}\left(0\right)=1,f^{\text{'}\text{'}}\left(0\right)=0,f^{\text{'}\text{'}\text{'}}\left(0\right)=-1$

  • 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 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 $f\left(x\right)=\sin x$

    • $\begin{array}{rcl}f\left(x\right)& =& f\left(0\right)+x{f}^{\text{'}}\left(0\right)+\frac{x^2}{2!}{f}^{\text{'}\text{'}}\left(0\right)+\frac{x^3}{3!}{f}^{\text{'}\text{'}\text{'}}\left(0\right)+...\\ & =& 0+x\left(1\right)+\frac{x^2}{2!}\left(0\right)+\frac{x^3}{3!}(-1)+...\end{array}$

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

  • E.g. for $f\left(x\right)=\sin x$

    • $\begin{array}{rcl}f\left(x\right)& =& x-\frac{x^3}{6}+...\end{array}$

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:

${\mathrm{e}}^x=1+x+\frac{x^2}{2!}+...$

$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-...$

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...$

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...$

$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-...$

• 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 ${(1+x)}^n$ 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