Definition: Taylor Series

Let $f$ be a function that has derivatives of all orders at $x = c$ (i.e. $f$ is infinitely differentiable, or smooth, at $c$ ). The Taylor series of $f$ centered at $c$ is the power series

17 viewsMathematics · Single Variable Calculus

Definition: Taylor Series

Let $f$ be a function that has derivatives of all orders at $x = c$ (i.e. $f$ is infinitely differentiable, or smooth, at $c$ ). The Taylor series of $f$ centered at $c$ is the power series

\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots,

where $f^{(n)}(c)$ denotes the $n$ -th derivative of $f$ evaluated at $c$ (with $f^{(0)}(c) = f(c)$ and $0! = 1$ ).

The coefficients $c_n = f^{(n)}(c)/n!$ are called the Taylor coefficients of $f$ at $c$ .

17 viewsMathematics · Single Variable Calculus

Definition: Taylor Series

Let $f$ be a function that has derivatives of all orders at $x = c$ (i.e. $f$ is infinitely differentiable, or smooth, at $c$ ). The Taylor series of $f$ centered at $c$ is the power series

17 viewsMathematics · Single Variable Calculus

\sum_{n = 0}^{\infty}

\frac{f^{(n)} (c)}{n!}

(

x

-

c

)^{n}

=

f

(

c

)

+

f^{'}

(

c

)

(

x

-

c

)

+

\frac{f^{''} (c)}{2!}

(

x

-

c

)^{2}

+

\frac{f^{'''} (c)}{3!}

(

x

-

c

)^{3}

+

\dots

,

\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots,