Taylor Series

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function’s derivatives at a single point.

The Formula

If a function $f(x)$ is infinitely differentiable at a real or complex number $a$, then the Taylor series for $f$ centered at $a$ is:

$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

Where:

• $f^{(n)}(a)$ denotes the $n$-th derivative of $f$ evaluated at $a$.
• $n!$ is the factorial of $n$.

Maclaurin Series

A Maclaurin series is simply a Taylor series centered at $a=0$:

$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n$

Common Examples

Function	Maclaurin Series	Interval of Convergence
$e^x$	${\sum }_{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\dots$	$(-\infty ,\infty )$
$\sin (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\dots$	$(-\infty ,\infty )$
$\cos (x)$	${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\dots$	$(-\infty ,\infty )$

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Draft note: Discuss Taylor’s Theorem and the Remainder Term (Lagrange error bound) in next iteration.