Taylor series, derivation of

Let $f(x)$ be given by the following power series:

$f(x)=c_0+c_1(x-a)+c_2{(x-a)}^2+\mathrm{\cdots }+c_n{(x-a)}^n+\mathrm{\cdots }={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{c}_k{(x-a)}^k$

Now let’s compute a few derivatives at $x=a.$

$f(a)=c_0;f^{\prime }(a)=c_1;f^{\prime \prime }(a)=2c_2;f^{(3)}(a)=6c_3;f^{(n)}(a)=n!c_n$

From this, it is clear that $c_n={\displaystyle \frac{f^{(n)}(a)}{n!}}$, thus the series can be written as:

$T_n={\displaystyle \sum _{k=0}^n}{c}_k{(x-a)}^k={\displaystyle \sum _{k=0}^n}{\displaystyle \frac{f^{(k)}(a)}{k!}}{(x-a)}^k$

where $f(x)=T_{\mathrm{\infty }}$.