Taylor Series

Topics: Calculus - Taylor Series

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(definition)

Let $f\in C_{\infty }[a,b]$ be function that’s differentiable $n$ times $\forall n\in \mathbb{N}$.

We define the Taylor series of $f$ around $c\in [a,b]$ as:

$$\sum _{n=1}^{\infty }\frac{f^n(c)(x-c{)}^n}{n!}$$

The Taylor series of $f$ equals $f$ in all of its domain.

Notice the Taylor series can be seen as a Taylor polynomial of infinite degree.

Example

For instance, let $f(x)=e^x$. The Taylor series of $f$ around $0$ is rather easy to calculate, since the derivative of $e^x$ is always itself, and $e^0=1$.

The Taylor series of this function is:

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots +\frac{x^k}{k!}+\dots$$