Regarding Taylor's Series theorem, i cannot prove this
Determine the Taylor series $T_f$ of $f(x) = \exp{x}$ about an arbitrary $a \in \mathbb{R}$ and show that it is identical to $f(x)$ for all $x \in \mathbb{R}$.
i also may use the hint that exp(x) = exp(a)*exp(x − a)
any ideas are appreciated !
I guess by its very definition of Taylor expansion you have $$ T_f(x)=e^a+\frac{e^a}{1}(x-a)+\frac{e^a}{2}(x-a)^2+\dots, $$ or more precisely $$T_f(x) = \sum\limits_{k=0}^\infty \frac{e^a}{k!}(x-a)^k=e^a \sum\limits_{k=0}^\infty \frac{(x-a)^k}{k!}=e^ae^{x-a}=e^x,$$ where I used the Definition of $\exp$. Maybe its formally better to write $exp(a)$ instead of $e^a$, depending on your definition.