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

$\displaystyle f(x)=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}+\cdots+c_{n}(x-a)^{n}+\cdots=\sum_{{k=0}}^{{\infty}}c_{k}(x-a)^{k}$

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

$\displaystyle f(a)=c_{0};\quad f^{{\prime}}(a)=c_{1};\quad f^{{\prime\prime}}(a)=2c_{2};\quad f^{{(3)}}(a)=6c_{3};\quad f^{{(n)}}(a)=n!c_{n}$

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

$\displaystyle T_{n}=\sum_{{k=0}}^{n}c_{k}(x-a)^{k}=\sum_{{k=0}}^{n}\frac{f^{{(k)}}(a)}{k!}(x-a)^{k}$

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