Taylor or Maclaurin Series for a Function (College Board AP® Calculus BC): Revision Note

Taylor or Maclaurin series for a function

What is the Taylor series for a function?

• Let $f$ be a function

  • If the function exists at $x=a$

  • and if the function is infinitely differentiable at $x=a$

    • i.e. if the derivative $f^{\left(n\right)}\left(a\right)$ exists for all $n=1,2,3,4,...$

  • then the Taylor series for $f$ about $x=a$ is

$\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n=f\left(a\right)+f^{\text{'}}\left(a\right)\left(x-a\right)+\frac{f^{\text{'}\text{'}}\left(a\right)}{2!}{\left(x-a\right)}^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n+...$

• Note that

  • The Taylor series is a power series

  • The Taylor series can be seen as a Taylor polynomial that 'goes on forever'

    • I.e. the nth degree Taylor polynomial is a truncated Taylor series, which stops at a particular value of $n$

    • A Taylor polynomial for $f$ is a partial sum of the Taylor series for $f$

What is the relationship between Taylor series and power series?

• A power series about $a$ is any series of the form

  • $\sum _{n=0}^{\infty }{c}_n{\left(x-a\right)}^n=c_0+c_1\left(x-a\right)+c_2{\left(x-a\right)}^2+...+c_n{\left(x-a\right)}^n+...$

    • where the coefficients $c_n$ can be any sequence of real numbers

• A Taylor series about $a$ is the series

  • $\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n=f\left(a\right)+f^{\text{'}}\left(a\right)\left(x-a\right)+\frac{f^{\text{'}\text{'}}\left(a\right)}{2!}{\left(x-a\right)}^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n+...$

    • where the coefficients must have the specific form $\frac{f^{\left(n\right)}\left(a\right)}{n!}$

• All Taylor series about $a$ for the function $f\left(x\right)$ are power series

  • As a Taylor series is a power series, it may converge

    • at a single point (i.e. at $x=a$ only)

    • over a finite interval of $x$ values

    • or for all real numbers

  • On its interval of convergence, a Taylor series is exactly equal to its function

    • It only becomes an approximation if it is truncated at some value of $n$

• Not all power series about $a$ are Taylor series for the function $f\left(x\right)$

  • If, however, a power series about $x=a$ is found such that it converges to the function $f\left(x\right)$ on some interval of convergence (i.e. has a positive radius of convergence), then that power series is the Taylor series for $f\left(x\right)$ about $a$

  • This means that the Taylor series for a function is unique

    • I.e., there is no other power series that will also be exactly equal to the function

What is the Maclaurin series for a function?

• A Maclaurin series for a function is a special case of a Taylor series

• It is the Taylor series for a function about the point $x=0$

  • I.e. if the function $f$ and all its derivatives exist at $x=0$, then the Maclaurin series for $f$ about $x=0$ is

  $\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(0\right)}{n!}{x}^n=f\left(0\right)+f^{\text{'}}\left(0\right)x+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}{x}^2+...+\frac{f^{\left(n\right)}\left(0\right)}{n!}{x}^n+...$

What standard power series should I be familiar with?

• You should know the following four Maclaurin series for standard functions

  • These are used as the basis for constructing the Maclaurin series for other functions

  • See the 'Representing Functions as Power Series' study guide

• $\frac{1}{1-x}=\sum _{n=0}^{\infty }{x}^n=1+x+x^2+x^3+...$

  • This series converges for $\left|x\right|<1$, i.e. $-1<x<1$

  • Note that this is a geometric series with common ratio $x$

    • Remember that the geometric series sum is $\sum _{n=0}^{\infty }a{r}^n=\frac{a}{1-r}$ for $\left|r\right|<1$

    • The power series version can be found by

      • dividing both sides of that by $a$

      • and substituting $r=x$

• $\sin x=\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^{2n+1}}{\left(2n+1\right)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$

  • This series converges for all real numbers, i.e. for $-\infty <x<\infty$

  • Note that this is an alternating series and $x$ is in radians

• $\cos x=\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^{2n}}{\left(2n\right)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$

  • This series converges for all real numbers, i.e. for $-\infty <x<\infty$

  • Note that this is an alternating series and $x$ is in radians

• $e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$

  • This series converges for all real numbers, i.e. $-\infty <x<\infty$

• These results are summarized in the table below

Do all Taylor series have an infinite number of non-zero terms?

• For some functions, the derivatives all become zero after a certain point

  • In particular, this is true of polynomial functions

• For example, if you try to calculate the Taylor series for $f\left(x\right)=x^2$ about $x=a$

  • The derivatives are $f^{\text{'}}\left(x\right)=2x$, $f^{\text{'}\text{'}}\left(x\right)=2$, $f^{\text{'}\text{'}\text{'}}\left(x\right)=0$, $f^{\left(4\right)}\left(x\right)=0$, $f^{\left(5\right)}\left(x\right)=0$, etc.

  • When you put that into the Taylor series formula you get

    $\begin{array}{rcl}\sum _{n=0}^{\infty }\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n& =& a^2+2a\left(x-a\right)+\frac{2}{2!}{\left(x-a\right)}^2+0+0+0+...\\ & =& a^2+2a\left(x-a\right)+{\left(x-a\right)}^2\end{array}$

    • Expanding gives$\begin{array}{rcl}& & a^2+2ax-2a^2+x^2-2ax+a^2\end{array}$ which simplifies to $x^2$

    • The Taylor series for $x^2$ is just $x^2$

• The Taylor series for a polynomial function is always just equal to the polynomial function

• For more 'interesting' functions, however, the Taylor series will generally go on forever