AP Calculus BC / Unit 10: Infinite Sequences and Series
10.14 Finding Taylor or Maclaurin Series for a Function
Notes
Know these Maclaurin series:
- $\displaystyle e^x=\sum\frac{x^n}{n!}$
- $\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$
- $\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$
- $\displaystyle \frac{1}{1-x}=1+x+x^2+\dots$ for $|x|<1$
Practice
The Maclaurin series for $\cos x$ has:
$\cos$ is an even function.
The $x^4$ coefficient for $e^x$:
$\displaystyle \frac{1}{4!}$.
The series $1-x+x^2-x^3+\dots$ represents:
Replace $x$ with $-x$ in $\displaystyle \frac{1}{1-x}$.
Ready for more? Take the Unit 10 test.