Taylor and Maclaurin Series

$\Sigma _{n = 0} ^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$, the Taylor series for $f$ centered at $a$

• $f^{(n)}(a)$ represents the $n$th derivative at $x =a$
• In the Maclaurin series, $a = 0$, our sum becomes $\Sigma _{n=0} ^\infty \frac{f^{(n)}(0)}{n!}x^n$

Taylor’s Remainder Theorem

Taylor's Remainder Theorem

$R_n(x) = \frac{f^{n+1}(z)(x-c)^{n+1}}{(n+1)!}$, where $z$ is the number that gives the largest value for the $n+1$th derivative of $f$

• $R_n$ is the exact value of this error
• $x-c$ gives the error
• $c$ is the value the polynomial is centered at
• $x$ is the value you wish to approximate
• Sometimes $x=z$
• Find the $n+1$th derivative and then find a $z$ value that maximizes it, and is between $c$ and $x$
  • This will give us the maximum error from the Taylor polynomial approximation