A Taylor Seeeerieeees

The Taylor series of a function $f(x)$ centered at $x = x_{0}$ is $\displaystyle\sum _{n = 0} ^{\infty} f^{(n)} (x_{0}) \frac{(x - x_{0}) ^{n}}{n!}$

In this case our function $f(x)$ is $e^{x}$ and $x_{0} = 2$ . The $n^{\text{th}}$ derivative of $e^x$ is $e^x$ itself, so $f^{(n)} (2) = e^2$ for all $n$ . Thus the Taylor series of $f(x) = e^x$ centered at $x = 2$ is $\sum _{n = 0} ^{\infty} e^2 \frac{(x - 2) ^{n}}{n!} \ _\square$