Answer
\begin{aligned}
T_{n}(x) =e+e(x-1)+\frac{e}{2}(x-1)^{2}+\frac{e}{6}(x-1)^{3}+\cdots+\frac{e}{n !}(x-1)^{n}
\end{aligned}
Work Step by Step
Since
\begin{array}{ll}
{f (x)=e^{x},} & {f (1)=e} \\
{f^{\prime}(x)=e^{x},} & {f^{\prime}(1)=e} \\ {f^{\prime \prime}(x)=e^{x},} & {f^{\prime \prime}(1)=e} \\ {f^{\prime \prime \prime}(x)=e^{x},} & {f^{\prime \prime \prime}(1)=e} \\ {\vdots} & {\vdots} \\ {f^{(n)}(x)=e^{x}} & {f^{(n)}(1)=e}\end{array}
Then
\begin{aligned}
T_{n}(x) &=f(1)+\frac{f^{\prime}(1)}{1 !}(x-1)+\frac{f^{\prime \prime}(1)}{2 !}(x-1)^{2}+\cdots+(x-1)^{n} \frac{f^{(n)}(1)}{n !} \\
&=e+e(x-1)+\frac{e}{2}(x-1)^{2}+\frac{e}{6}(x-1)^{3}+\cdots+\frac{e}{n !}(x-1)^{n}
\end{aligned}
